[{"data":1,"prerenderedAt":1219},["ShallowReactive",2],{"layer:lines:understand":3},{"layer":4,"contentHash":1198,"dependencyHashes":1199,"approval":1212,"releaseId":1218},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1193,"reviewStatus":1194,"authoring":1195},1,"lines","en","understand","Names, notation and rules for lines","Precise definitions, careful measuring and the mix-ups they clear up","Pin down point, line and plane; name lines, rays and segments correctly; measure without parallax error; and define collinear, concurrent, parallel and perpendicular lines precisely.",[13,14,15,16,17],"Use correct names and notation (written in words) for lines, rays, segments and lengths.","Explain why exactly one line passes through two points, and count lines through collinear and non-collinear points.","Measure and compare segments accurately with a ruler and a divider, avoiding parallax error.","Define intersecting, concurrent, parallel and perpendicular lines, transversals and the perpendicular bisector.","Use AB + BC = AC for points on a segment, and spot when it cannot hold.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Line, ray, segment by sight (Discover)",{"label":32,"value":33},"Chapters","12",{"label":35,"value":36},"Labs","Notation match, collinear sort, line spotter, property sort",{"label":38,"value":39},"Symbols","∥ parallel, ⊥ perpendicular",[41,45,51,57,60,77,82,85,90,93,128,133,148,178,183,186,191,196,211,216,219,223,248,271,276,281,284,300,309,319,334,348,353,356,385,388,393,417,425,429,441,446,449,453,456,510,514,523,541,546,549,553,573,597,602,607,610,628,631,635,639,645,650,653,669,672,683,688,697,706,726,741,745,750,753,763,766,790,802,806,814,835,844,849,857,893,952,1017,1153,1157,1173,1177,1181],{"id":42,"type":43,"markdown":44},"intro-understand","prose","In Discover you met points, lines, rays and segments by looking at torch beams and railway tracks. Now we tighten everything up. What exactly counts as a line? How do you **name** one so that nobody can misunderstand you? How many lines can pass through two points, or three? How do you **measure** a segment accurately, and what goes wrong when you do not? And what precisely makes two lines parallel or perpendicular?\n\nThese are the questions a Class 6 or 7 textbook answers, and they are the foundations of every angle, triangle and construction you will ever meet. Getting them exactly right now saves a lot of confusion later.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how-u","callout","observation","How to use this lesson","Read the chapters in order the first time: each one uses the words from the one before. Work every **worked example** with a pencil before reading the steps. The two games in the middle are for practice, so play them until you can answer without hesitating.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","Where geometry starts: point, line, plane","Chapter 01","1 Starting ideas",{"id":58,"type":43,"markdown":59},"undefined","Try to define a point. \"A small dot\"? But a dot has a size, and a point has none. \"A position\"? Then what is a position? Every definition uses other words, which need their own definitions, and so on forever.\n\nMathematicians solved this more than 2,000 years ago by **not defining** a few starting ideas. They simply describe them clearly and agree to use them. In school geometry the three starting ideas are the **point**, the **line** and the **plane**. Everything else, such as a segment, a ray, an angle or a triangle, is then defined using these three.",{"id":61,"type":62,"tone":63,"items":64},"spec-undefined","spec","blue",[65,69,73],{"label":66,"big":67,"value":68},"Point","no size","Marks a position. No length, width or thickness.",{"label":70,"big":71,"value":72},"Line","no ends","Perfectly straight, no thickness, goes on forever both ways.",{"label":74,"big":75,"value":76},"Plane","no edges","Perfectly flat, no thickness, goes on forever in every direction.",{"id":78,"type":47,"variant":79,"title":80,"markdown":81},"nuance-pictures","nuance","Every drawing is only a picture","A pencil dot is about half a millimetre wide. A pencil line is about half a millimetre thick. A page stops at its edges. So we can never *draw* a true point, line or plane. We draw **pictures** that remind us of them, and we reason about the perfect objects in our heads. That is why a careful proof beats a careful drawing: drawings can mislead, reasoning cannot (if it is correct).",{"id":83,"type":43,"markdown":84},"thin-thought","Here is a way to feel why a point has no size. Take a segment 1 cm long and cut it in half: 0.5 cm. Halve again: 0.25 cm. Keep halving. After 10 halvings the piece is 1 ÷ 1,024 cm, about one hundredth of a millimetre. After 20 halvings it is about one millionth of a centimetre, far thinner than a hair. You could go on forever and never get down to zero. A point is what is left at the very end of that endless shrinking: position, but no size.",{"id":86,"type":53,"title":87,"eyebrow":88,"navLabel":89},"ch02","Naming things so nobody gets confused","Chapter 02","2 Naming and notation",{"id":91,"type":43,"markdown":92},"naming-rules","Geometry is written in a very compact language. The rules are simple:\n\n- **Points** get **capital letters**: A, B, P, Q.\n- **Lines** can be named by **two points on them** (line AB) or by a **small letter** (line *l*, line *m*).\n- **Segments** are named by their **two end points** (segment AB).\n- **Rays** are named by their **end point first**, then any other point on the ray (ray AB).\n\nTextbooks also use small symbols drawn **over** the letters. Because those symbols are hard to type, this site writes the words instead. Here is what each symbol looks like, described in words.",{"id":94,"type":95,"caption":96,"columns":97,"rows":102},"table-notation","table","How textbooks write it, and how we write it here",[98,99,100,101],"Object","Textbook symbol (in words)","Written on this site","Does the order of letters matter?",[103,108,113,118,123],[104,105,106,107],"Line through A and B","AB with a bar over it that has an arrowhead at **both** ends","line AB","No: line AB = line BA",[109,110,111,112],"Ray from A through B","AB with a bar over it that has an arrowhead at the **right** end only","ray AB","Yes: ray AB ≠ ray BA",[114,115,116,117],"Segment from A to B","AB with a plain bar over it (no arrowheads)","segment AB","No: segment AB = segment BA",[119,120,121,122],"Length of segment AB","AB with nothing over it","AB = 5 cm","No",[124,125,126,127],"Line named by a letter","a small italic letter","line l","Not applicable",{"id":129,"type":47,"variant":130,"title":131,"markdown":132},"careful-length","careful","segment AB is a thing; AB is a number","When you write **AB = 5 cm**, AB (with no bar) stands for the **length** of the segment, which is a number. When you write **segment AB**, you mean the segment itself, the actual set of points. It is correct to say \"segment AB has length 5 cm\" or \"AB = 5 cm\". It is not correct to say \"line AB = 5 cm\": a line has no length.",{"id":134,"type":135,"title":136,"problem":137,"steps":138,"help":146},"we-names","worked_example","How many names does one line have?","Three points P, Q and R lie on one straight line, in that order. List every way of naming the line using two of the points. Then list the segments and the rays (starting at one marked point and passing through another).",[139,140,141,142,143,144,145],"**Line names.** Choose any two of the three points, in either order: line PQ, QP, PR, RP, QR, RQ. That is **6 names**, but they all name the **same line**.","**Segments.** A segment is fixed by its two end points, and order does not matter: segment PQ, segment QR, segment PR. That is **3 different segments**.","Check the lengths fit together: since Q lies between P and R, PQ + QR = PR.","**Rays from P:** ray PQ and ray PR both start at P and head the same way, so they are the **same** ray. That is 1 ray.","**Rays from Q:** ray QP heads towards P, ray QR heads towards R. Opposite directions, so **2** rays.","**Rays from R:** ray RQ and ray RP are the same ray. 1 ray.","Total: 1 + 2 + 1 = **4 different rays**, although you can write 6 ray names.",{"simplerExplanation":147},"Different names can describe the same thing, just as \"Mumbai\" and \"Bombay\" name the same city. For rays, check where it starts and which way it points.",{"id":149,"type":150,"component":151,"componentVersion":5,"config":152,"objective":176,"textAlternative":177},"lab-match-notation","interactive","match-pairs",{"prompt":153,"mode":154,"pairs":155},"Match each piece of notation (written in words) to what it means.","connect",[156,158,160,163,165,168,171,174],{"a":106,"b":157},"Goes on forever through A and B, both ways",{"a":111,"b":159},"Starts at A, goes on forever through B",{"a":161,"b":162},"ray BA","Starts at B, goes on forever through A",{"a":116,"b":164},"The straight path from A to B, stopping at both",{"a":166,"b":167},"AB = 4 cm","The length of segment AB is 4 cm",{"a":169,"b":170},"AB ∥ CD","Line AB is parallel to line CD",{"a":172,"b":173},"AB ⊥ CD","Line AB is perpendicular to line CD",{"a":126,"b":175},"A line named with a small letter","Connect each piece of geometry notation to its meaning.","Connect each piece of notation on the left with its meaning on the right.\n\n- **line AB:** goes on forever through A and B, in both directions.\n- **ray AB:** starts at A and goes on forever through B.\n- **ray BA:** starts at B and goes on forever through A (the opposite direction).\n- **segment AB:** the straight path from A to B, stopping at both.\n- **AB = 4 cm:** the length of segment AB is 4 cm.\n- **AB ∥ CD:** line AB is parallel to line CD.\n- **AB ⊥ CD:** line AB is perpendicular to line CD.\n- **line l:** a line named with a small letter instead of two points.\n\nThe pair people mix up most is ray AB and ray BA: the first letter is always where the ray starts.",{"id":179,"type":53,"title":180,"eyebrow":181,"navLabel":182},"ch03","How many lines through a point, or two?","Chapter 03","3 Lines through points",{"id":184,"type":43,"markdown":185},"through-one","Mark a single point A on a page. Draw a straight line through it. Now draw another, at a different slant. And another. You can keep going forever: **through one point there are endlessly many lines**, one for every direction.\n\nNow mark two points, A and B. Try to draw two *different* straight lines that both pass through A and B. You cannot. As soon as a straight line passes through A and B, its direction is fixed. **Through two different points there is exactly one line.**",{"id":187,"type":47,"variant":188,"title":189,"markdown":190},"def-two-points","definition","Two points fix a line","Through any two different points there passes **one and only one** straight line. This is so basic that Euclid made it the very first of his starting rules (postulates). It is why we can name a line by any two of its points.",{"id":192,"type":47,"variant":193,"title":194,"markdown":195},"example-chalk","example","The mason's chalk line","Builders and tailors in India have used this fact for centuries. A mason who wants a perfectly straight mark on a wall or floor rubs a string with chalk or red powder, pins the two ends at two points and plucks the string like a guitar. It snaps against the surface and leaves a straight line. Two fixed points are all that is needed; the string finds the only straight line through them. A tailor's chalk and a stretched measuring tape do the same on cloth.",{"id":197,"type":198,"prompt":199,"options":200,"explanation":210},"predict-bend","prediction","You have fixed a stretched string at two nails A and B. Can you move the middle of the string sideways and still keep it straight and passing through both nails?",[201,204,207],{"id":202,"label":203},"a","Yes, it can swing to the left or right",{"id":205,"label":206},"b","No, there is only one straight path through A and B",{"id":208,"label":209},"c","Only if the nails are far apart","**No.** Once two points are fixed, there is only one straight line through them. Pushing the middle sideways makes the string bend, so it is no longer straight. This is exactly why a carpenter needs only two nails to fix a straight batten, but a single nail lets it spin round.",{"id":212,"type":53,"title":213,"eyebrow":214,"navLabel":215},"ch04","Rays in detail","Chapter 04","4 Rays",{"id":217,"type":43,"markdown":218},"ray-detail","A ray has one end point and goes on forever in one direction. To name it, **always write the end point first**. Ray AB starts at A and goes through B and beyond. It includes the point A itself.\n\nIf point C lies on ray AB beyond B, then ray AB and ray AC are the **same ray**: same start, same direction. Changing the second letter to another point on the ray does not change the ray. Changing the **first** letter always does.",{"id":220,"type":47,"variant":188,"title":221,"markdown":222},"def-opposite","Opposite rays","Two rays that share the **same end point** and point in **exactly opposite directions** are called **opposite rays**. Together they make a whole line. If Q lies between P and R on a line, then ray QP and ray QR are opposite rays.",{"id":224,"type":95,"caption":225,"columns":226,"rows":231},"table-ray-segment-line","Comparing the three, precisely",[227,228,229,230],"Property","Segment AB","Ray AB","Line AB",[232,237,240,242,244,246],[233,234,235,236],"End points","A and B","A only","None",[238,239,239,239],"Contains A?","Yes",[241,122,239,239],"Contains points beyond B?",[243,122,122,239],"Contains points beyond A (on the far side from B)?",[245,239,122,122],"Has a length?",[247,239,122,239],"Same as the BA version?",{"id":249,"type":250,"itemId":251,"prompt":252,"check":253,"hints":266,"feedback":268},"prac-ray-same","practice","lines.understand-ray-same","Points X, Y and Z lie on a line in that order. Which rays are the same as ray XY?",{"kind":254,"options":255,"correct":265},"choice",[256,258,260,262],{"id":202,"label":257},"Ray YX",{"id":205,"label":259},"Ray XZ",{"id":208,"label":261},"Ray ZX",{"id":263,"label":264},"d","Ray YZ",[205],[267],"The ray must start at X and head the same way as Y.",{"correct":269,"incorrect":270},"Right: ray XZ also starts at X and heads through Y towards Z.","Ray XY starts at X and heads towards Y and Z. **Ray XZ** starts at the same place and heads the same way, so it is the same ray. The others start at a different point.",{"id":272,"type":47,"variant":273,"title":274,"markdown":275},"mis-ray-direction","misconception","“Ray AB and ray BA are the same ray”","They share the segment AB, but ray AB carries on past B and ray BA carries on past A. They start at different points and point in opposite directions. The only points they have in common are those of segment AB.",{"id":277,"type":53,"title":278,"eyebrow":279,"navLabel":280},"ch05","Line segments and their lengths","Chapter 05","5 Segments and length",{"id":282,"type":43,"markdown":283},"seg-detail","A segment is the only one of the three with a **length**, so it is the one you can measure, compare, add and subtract.\n\nIf point B lies **on** segment AC, somewhere between A and C, then the two smaller segments fit together exactly: **AB + BC = AC**. This \"betweenness\" rule is behind many textbook problems. If B is **not** on segment AC, then AB + BC is **more** than AC, because the straight path is the shortest.",{"id":285,"type":286,"items":287},"formulas-seg","formulas",[288,291,294,297],{"expression":289,"caption":290},"AB + BC = AC","When B lies on segment AC, between A and C.",{"expression":292,"caption":293},"AB + BC > AC","When B is not on segment AC: going via B is a detour.",{"expression":295,"caption":296},"AM = MB = AB ÷ 2","M is the midpoint of AB: it cuts the segment into two equal halves.",{"expression":298,"caption":299},"1 cm = 10 mm","Ruler markings: each centimetre has ten millimetre divisions.",{"id":301,"type":135,"title":302,"problem":303,"steps":304},"we-between","Finding a missing piece","Points A, B and C lie on a line with B between A and C. AC = 11.2 cm and AB = 4.7 cm. Find BC.",[305,306,307,308],"B is between A and C, so AB + BC = AC.","4.7 + BC = 11.2.","BC = 11.2 − 4.7 = **6.5 cm**.","Check: 4.7 + 6.5 = 11.2 ✓",{"id":310,"type":135,"title":311,"problem":312,"steps":313},"we-midpoint","The midpoint","M is the midpoint of segment PQ and PM = 3.6 cm. How long is PQ? If R is the midpoint of PM, how long is RQ?",[314,315,316,317,318],"M is the midpoint, so MQ = PM = 3.6 cm.","PQ = PM + MQ = 3.6 + 3.6 = **7.2 cm**.","R is the midpoint of PM, so PR = RM = 3.6 ÷ 2 = 1.8 cm.","RQ = RM + MQ = 1.8 + 3.6 = **5.4 cm**.","Check: PR + RQ = 1.8 + 5.4 = 7.2 = PQ ✓",{"id":320,"type":250,"itemId":321,"prompt":322,"check":323,"hints":328,"feedback":331},"prac-between","lines.understand-between","L, M and N lie on a line with M between L and N. LM = 38 mm and MN = 5.4 cm. How long is LN, in cm?",{"kind":324,"answer":325,"tolerance":326,"unit":327},"number",9.2,0,"cm",[329,330],"Put both lengths in the same unit first.","38 mm = 3.8 cm.",{"correct":332,"incorrect":333},"Yes: 3.8 + 5.4 = 9.2 cm.","Convert first: 38 mm = 3.8 cm. Then LN = LM + MN = 3.8 + 5.4 = **9.2 cm**.",{"id":335,"type":250,"itemId":336,"prompt":337,"check":338,"hints":343,"feedback":345},"prac-not-between","lines.understand-not-between","PQ = 5 cm, QR = 3 cm and PR = 7 cm. Can Q lie between P and R on segment PR?",{"kind":254,"options":339,"correct":342},[340,341],{"id":202,"label":239},{"id":205,"label":122},[205],[344],"If Q were between P and R, what would PQ + QR equal?",{"correct":346,"incorrect":347},"Right: 5 + 3 = 8, not 7, so Q is not on segment PR.","If Q were on segment PR we would have PQ + QR = PR. But 5 + 3 = 8 ≠ 7, so **Q is not between P and R**: the three points are not even on one line.",{"id":349,"type":53,"title":350,"eyebrow":351,"navLabel":352},"ch06","Measuring and comparing segments","Chapter 06","6 Measuring",{"id":354,"type":43,"markdown":355},"compare-three","There are three ways to compare two segments, and they get more reliable as you go:\n\n1. **By observation**, just looking. Quick, but easily fooled, especially when the segments point in different directions or have decorations at the ends.\n2. **By tracing**, copying one segment onto thin paper and laying it on the other. Better, but clumsy and not very precise.\n3. **By measuring** with a ruler, or with a **divider** and a ruler. This gives an actual number you can write down and check.",{"id":357,"type":95,"caption":358,"columns":359,"rows":364},"table-compare","Three ways to compare two segments",[360,361,362,363],"Method","How","Good for","Weakness",[365,370,375,380],[366,367,368,369],"Observation","Look at both and judge","Very different lengths","Optical illusions; nearly equal lengths",[371,372,373,374],"Tracing","Copy one on tracing paper, lay it on the other","Checking equal lengths without numbers","Slow; paper slips; no number",[376,377,378,379],"Ruler","Read both ends on the scale and subtract","Any segment that fits the ruler","Parallax error; thick ruler edge",[381,382,383,384],"Divider + ruler","Open the divider to the segment, then place it on the ruler","Accurate readings, curved rulers, small segments","Needs a steady hand",{"id":386,"type":43,"markdown":387},"parallax","Rulers have thickness. The scale is printed on top of the plastic, but the segment is on the page underneath. If you look at the ruler **from the side**, the mark you see lined up with the end of the segment is not the right one: it shifts, and your reading is off by a millimetre or more. This is called **parallax error**.\n\nThe cure is simple: put your eye **directly above** the point you are reading. Even better, stand the ruler on its edge so that the markings touch the paper, or use a divider, which touches the paper with sharp points.",{"id":389,"type":47,"variant":390,"title":391,"markdown":392},"tryit-parallax","try_it","See parallax with your own finger","Hold one finger up at arm's length in front of a door frame. Close your left eye, then open it and close your right eye. Your finger seems to jump sideways against the door frame, although it did not move. Your two eyes look from slightly different places, so they line it up with different parts of the background. The same thing happens when you read a ruler from the side.",{"id":394,"type":395,"title":396,"items":397},"steps-divider","steps","Measuring a segment with a divider",[398,402,405,409,413],{"title":399,"tag":400,"text":401},"Open the divider","tool from the geometry box","A divider looks like a compass with two sharp points and no pencil.",{"title":403,"tag":234,"text":404},"Fit the ends","Place one point exactly on A and open the arms until the other point is exactly on B.",{"title":406,"tag":407,"text":408},"Lock it","do not squeeze","Lift the divider carefully so the opening does not change.",{"title":410,"tag":411,"text":412},"Move to the ruler","one point on 0","Put one point on the 0 mark of the ruler (or on 1 cm if the 0 is worn).",{"title":414,"tag":415,"text":416},"Read the other point","eye straight above","Read where the second point lands. If you started at 1 cm, subtract 1.",{"id":418,"type":135,"title":419,"problem":420,"steps":421},"we-broken-ruler","The broken-ruler method","Kabir's ruler is chipped and the first 1 cm is missing. He lines one end of a segment up with the 3 cm mark and the other end falls on the 10.4 cm mark. How long is the segment?",[422,423,424],"The length is the **distance between** the two readings, not the reading at the far end.","10.4 − 3 = **7.4 cm**, which is 74 mm.","Starting from a whole-centimetre mark such as 3 cm makes the subtraction easy and avoids the worn end.",{"id":426,"type":47,"variant":273,"title":427,"markdown":428},"mis-eye","“You can tell which is longer just by looking”","Draw two equal segments. Add arrowheads pointing **outward** at the ends of one, like this: \u003C >, and pointing **inward** at the ends of the other: > \u003C. Most people now say the second one is longer. It is not. This famous trick is the Müller-Lyer illusion. Measure before you trust your eyes.",{"id":430,"type":250,"itemId":431,"prompt":432,"check":433,"hints":436,"feedback":438},"prac-parallax","lines.understand-ruler-read","A segment starts at the 1.5 cm mark on a ruler and ends at the 8.2 cm mark. What is its length in millimetres?",{"kind":324,"answer":434,"tolerance":326,"unit":435},67,"mm",[437],"Find the length in cm first, then multiply by 10.",{"correct":439,"incorrect":440},"Correct: 8.2 − 1.5 = 6.7 cm = 67 mm.","Length = 8.2 − 1.5 = 6.7 cm. Then 6.7 × 10 = **67 mm**.",{"id":442,"type":53,"title":443,"eyebrow":444,"navLabel":445},"ch07","Collinear and non-collinear points","Chapter 07","7 Collinear points",{"id":447,"type":43,"markdown":448},"collinear","Three or more points that all lie on **one straight line** are called **collinear** (\"co\" means together, \"linear\" means on a line). If no single straight line passes through all of them, they are **non-collinear**.\n\nAny **two** points are always collinear, because a line passes through any two points. The question only becomes interesting with **three or more** points.",{"id":450,"type":47,"variant":188,"title":451,"markdown":452},"def-collinear","Collinear points","Points are **collinear** if one straight line passes through all of them, and **non-collinear** if no such line exists. Three non-collinear points are the three corners of a triangle.",{"id":454,"type":43,"markdown":455},"three-points-lines","How many lines can you draw through pairs of three points? It depends on whether they are collinear:\n\n- **Collinear** A, B, C: lines AB, BC and AC are all the **same** line. Only **1** line.\n- **Non-collinear** A, B, C: lines AB, BC and CA are all different. **3** lines, forming the sides of a triangle.\n\nNo other answer is possible: either all three points share one line, or no two of the three lines coincide.",{"id":457,"type":150,"component":458,"componentVersion":5,"config":459,"objective":508,"textAlternative":509},"lab-sort-collinear","sort-game",{"prompt":460,"bins":461,"items":467,"seconds":326},"Are these points collinear (on one straight line) or not?",[462,464],{"id":447,"label":463},"Collinear",{"id":465,"label":466},"noncollinear","Non-collinear",[468,472,476,480,484,488,492,496,500,504],{"id":469,"label":470,"bin":447,"why":471},"beads","Three beads on a tightly stretched thread","A tight thread is straight, so the beads lie on one line.",{"id":473,"label":474,"bin":465,"why":475},"triangle","The three corners of a triangle","If they were on one line there would be no triangle.",{"id":477,"label":478,"bin":447,"why":479},"stumps","The tops of the three stumps at one end of a cricket pitch","The stumps stand in a straight row.",{"id":481,"label":482,"bin":447,"why":483},"clock6","The centre of a clock and the tips of both hands at 6:00","The hands point straight up and straight down: one line.",{"id":485,"label":486,"bin":465,"why":487},"clock3","The centre of a clock and the tips of both hands at 3:00","The hands are at a right angle, so the three points make a corner.",{"id":489,"label":490,"bin":447,"why":491},"pole-tops","Tops of electric poles along a straight, level road","Equal poles on a straight level road line up.",{"id":493,"label":494,"bin":465,"why":495},"square-corners","Three corners of a square","They make a right-angled triangle.",{"id":497,"label":498,"bin":447,"why":499},"ruler-marks","The 2 cm, 5 cm and 9 cm marks on a ruler","All ruler marks lie along one straight edge.",{"id":501,"label":502,"bin":465,"why":503},"stars","Three stars that make a small triangle in the sky","They form a triangle, not a straight row.",{"id":505,"label":506,"bin":447,"why":507},"midpoint","The two ends of a segment and its midpoint","The midpoint lies on the segment itself.","Decide whether sets of three points are collinear or non-collinear.","This game gives ten sets of points. Decide whether each set lies on one straight line.\n\n**Collinear:** three beads on a tight thread; the tops of the three stumps at one end of a cricket pitch; the centre of a clock and both hand tips at 6:00 (the hands point straight up and down); the tops of equal poles on a straight level road; the 2, 5 and 9 cm marks on a ruler; the ends of a segment and its midpoint.\n\n**Non-collinear:** the corners of a triangle; the centre of a clock and both hand tips at 3:00 (the hands make a right angle); three corners of a square; three stars forming a small triangle.\n\nA quick test on paper: lay a ruler through two of the points and see whether the third touches the edge.",{"id":511,"type":47,"variant":193,"title":512,"markdown":513},"example-check-collinear","Checking collinearity with lengths","Measure all three distances. If the two shorter ones add up **exactly** to the longest, the points are collinear. For example, if AB = 3 cm, BC = 5 cm and AC = 8 cm, then 3 + 5 = 8, so B lies on segment AC and the points are collinear. If AC were 7 cm, they could not be, because 3 + 5 ≠ 7.",{"id":515,"type":135,"title":516,"problem":517,"steps":518},"we-collinear-check","Are they collinear?","Three points have these distances between them: PQ = 4.6 cm, QR = 3.9 cm and PR = 8.5 cm. A second set has XY = 5 cm, YZ = 6 cm and XZ = 10 cm. Which set is collinear?",[519,520,521,522],"For collinear points, the two shorter distances add up exactly to the longest.","First set: 4.6 + 3.9 = 8.5 = PR. ✓ So Q lies on segment PR, and **P, Q, R are collinear**.","Second set: 5 + 6 = 11, but XZ = 10. Since 11 ≠ 10, **X, Y, Z are not collinear**: they form a triangle.","Notice that 11 is more than 10. The two shorter sides of a triangle always add up to more than the third side.",{"id":524,"type":250,"itemId":525,"prompt":526,"check":527,"hints":536,"feedback":538},"prac-collinear-lengths","lines.understand-collinear-lengths","AB = 7 cm, BC = 2.5 cm and AC = 4.5 cm. Which statement is true?",{"kind":254,"options":528,"correct":535},[529,531,533],{"id":202,"label":530},"The points are collinear, with C between A and B",{"id":205,"label":532},"The points are collinear, with B between A and C",{"id":208,"label":534},"The points are not collinear",[202],[537],"Which is the longest distance? Do the other two add up to it?",{"correct":539,"incorrect":540},"Yes: AC + CB = 4.5 + 2.5 = 7 = AB, so C lies between A and B.","The longest is AB = 7. The other two add up to 4.5 + 2.5 = 7, so the points are collinear, and the point not named in AB, which is **C**, lies between A and B.",{"id":542,"type":53,"title":543,"eyebrow":544,"navLabel":545},"ch08","Intersecting and concurrent lines","Chapter 08","8 Intersecting lines",{"id":547,"type":43,"markdown":548},"intersect-detail","Two lines in a plane that share a point are **intersecting lines**, and the shared point is their **point of intersection**. Why can they share only **one** point? Because if they shared two points, there would be two different lines through the same two points, and we know there is exactly one. So two distinct lines meet in **at most one** point.\n\nWhen **three or more** lines all pass through the **same** point, they are called **concurrent lines**, and the shared point is the **point of concurrence**. Think of the spokes of a bicycle wheel meeting at the hub, or the folds you get when you fold a paper circle in half several times.",{"id":550,"type":47,"variant":188,"title":551,"markdown":552},"def-concurrent","Concurrent lines","Three or more lines are **concurrent** if they all pass through one common point. Two lines that meet are just called intersecting; the word concurrent is used for three or more.",{"id":554,"type":250,"itemId":555,"prompt":556,"check":557,"hints":568,"feedback":570},"prac-concurrent","lines.understand-concurrent","Which of these is a picture of **concurrent** lines?",{"kind":254,"options":558,"correct":567},[559,561,563,565],{"id":202,"label":560},"The spokes of a bicycle wheel, extended into full lines",{"id":205,"label":562},"The rungs of a ladder",{"id":208,"label":564},"Two roads crossing at a chowk",{"id":263,"label":566},"The lines of a notebook",[202],[569],"Concurrent needs three or more lines through one point.",{"correct":571,"incorrect":572},"Yes: all the spokes pass through the hub.","Spokes all pass through the hub, so extended into lines they are **concurrent**. Two roads crossing are only two lines (intersecting); ladder rungs and notebook lines are parallel.",{"id":574,"type":95,"caption":575,"columns":576,"rows":580},"table-three-lines","How can three lines in a plane sit?",[577,578,579],"Arrangement","Points of intersection","Example",[581,585,589,593],[582,583,584],"All three parallel","0","Three rails of a ladder-shaped railing",[586,587,588],"All three through one point (concurrent)","1","Three spokes of a wheel extended into full lines",[590,591,592],"Two parallel, the third crossing both","2","A road crossing two parallel railway rails",[594,595,596],"No two parallel, not concurrent","3","Three roads forming a triangle of junctions",{"id":598,"type":47,"variant":599,"title":600,"markdown":601},"aha-at-most","aha","“At most one” is a strong statement","Notice the careful wording: two lines meet in **at most** one point. That allows two possibilities: exactly one point (intersecting) or no points (parallel). It rules out two, three or more. Mathematicians use \"at most\" and \"at least\" all the time to say exactly what is possible and what is not.",{"id":603,"type":53,"title":604,"eyebrow":605,"navLabel":606},"ch09","Parallel lines, precisely","Chapter 09","9 Parallel lines",{"id":608,"type":43,"markdown":609},"parallel-precise","In Discover we said parallel lines \"never meet\". The precise definition adds one important condition: **two lines are parallel if they lie in the same plane and never meet**, however far they are extended.\n\nWhy insist on \"the same plane\"? Look at the edge where your classroom's front wall meets the ceiling, and the edge where a side wall meets the floor. They never meet. But they are not parallel either: one runs across the room, the other runs along it, at different heights. Lines like this, in different planes, are called **skew lines**. You will meet them properly in the Extend layer.",{"id":611,"type":62,"tone":63,"items":612},"spec-parallel-facts",[613,617,621,624],{"label":614,"big":615,"value":616},"Symbol","∥","AB ∥ CD means line AB is parallel to line CD.",{"label":618,"big":619,"value":620},"Distance","constant","Measured along a perpendicular, the gap is the same everywhere.",{"label":622,"big":583,"value":623},"Meeting points","Parallel lines never share a point.",{"label":625,"big":626,"value":627},"On a page","same plane","Any two lines you draw on one page are in the same plane.",{"id":629,"type":43,"markdown":630},"parallel-distance","How do you measure the gap between two parallel lines? Not along any slanting segment you like: that would give different answers depending on the slant. The distance is measured along a segment that is **perpendicular** to both lines. That shortest gap is the same everywhere along parallel lines, which gives a practical test: measure the perpendicular gap at two places far apart. If the gaps are equal, the lines are parallel.",{"id":632,"type":47,"variant":273,"title":633,"markdown":634},"mis-short-lines","“They don't touch on the page, so they're parallel”","Two short segments can look separate on a page and still meet if you extend them far enough. A gap that narrows from 2.0 cm at one end to 1.8 cm at the other means the lines are slowly closing in and will meet somewhere off the page. Parallel is about the **whole lines**, not the pieces you happened to draw.",{"id":636,"type":47,"variant":188,"title":637,"markdown":638},"def-transversal","Transversal","A line that crosses **two or more** lines at **different points** is called a **transversal**. A road crossing both rails of a level crossing is a transversal of the rails. When a transversal cuts two lines it makes eight angles, which you will study in the Angles topic.",{"id":640,"type":641,"conceptId":642,"relation":643,"explanation":644},"conn-angles-transversal","connection","angles","helps_understand","A transversal crossing two parallel lines makes pairs of equal angles, the key to many angle problems.",{"id":646,"type":53,"title":647,"eyebrow":648,"navLabel":649},"ch10","Perpendicular lines and the perpendicular bisector","Chapter 10","10 Perpendicular lines",{"id":651,"type":43,"markdown":652},"perp-precise","Two lines are **perpendicular** if they intersect at a **right angle**, 90°. In a diagram a right angle is marked with a **small square** drawn in the corner, instead of the usual arc. We write AB ⊥ CD.\n\nWhen two lines are perpendicular, all **four** angles at the crossing are right angles. You only need to check one: if one angle is 90°, the one next to it along the straight line must be 180° − 90° = 90° too, and so on round.",{"id":654,"type":395,"title":655,"items":656},"steps-check-perp","Two ways to check a right angle",[657,661,665],{"title":658,"tag":659,"text":660},"Set square","from the geometry box","Place the square corner of a set square in the corner. If both edges lie exactly along both lines, the lines are perpendicular.",{"title":662,"tag":663,"text":664},"Folded paper","free and accurate","Fold any scrap of paper, then fold the crease onto itself. The new corner is exactly 90°. Use it like a set square.",{"title":666,"tag":667,"text":668},"Protractor","measure","Put the centre on the crossing and the base line on one line; the other should pass through 90.",{"id":670,"type":43,"markdown":671},"perp-bisector","A special perpendicular line is the **perpendicular bisector** of a segment. \"Bisect\" means cut into two equal parts. The perpendicular bisector of segment AB is the line that passes through the **midpoint** of AB **and** is perpendicular to it.\n\nYou can make one without any tools: draw segment AB on thin paper and fold the paper so that A lands exactly on B. Crease it. The crease passes through the midpoint (the two halves match) and meets AB at a right angle (the two angles at the fold match and add to 180°). Every point on that crease is the same distance from A as from B.",{"id":673,"type":250,"itemId":674,"prompt":675,"check":676,"hints":678,"feedback":680},"prac-perp-bisector","lines.understand-perp-bisector","Segment XY is 12 cm long. Its perpendicular bisector meets XY at M. Point K on the bisector is 10 cm from X. How far is K from Y, in cm?",{"kind":324,"answer":677,"tolerance":326,"unit":327},10,[679],"Every point on the perpendicular bisector is the same distance from both ends.",{"correct":681,"incorrect":682},"Right: KY = KX = 10 cm.","K is on the perpendicular bisector of XY, so it is equally far from X and Y: KY = KX = **10 cm**. (XM = MY = 6 cm, but that is not what was asked.)",{"id":684,"type":641,"conceptId":685,"relation":686,"explanation":687},"conn-constructing","constructing-angles","applied_in","The perpendicular bisector is constructed with a ruler and compass; its crossing with AB makes four right angles.",{"id":689,"type":135,"title":690,"problem":691,"steps":692},"we-perp-bisector","Using the perpendicular bisector","Segment AB is 9 cm long. Line l is its perpendicular bisector and meets AB at M. P is a point on l. PA = 7.5 cm. Find AM, MB and PB.",[693,694,695,696],"l passes through the midpoint M, so AM = MB = 9 ÷ 2 = **4.5 cm**.","Folding along l puts A exactly on B, and P stays where it is, because P lies on the fold.","So segment PA lands exactly on segment PB: PB = PA = **7.5 cm**.","Every point on the perpendicular bisector is the same distance from both ends of the segment.",{"id":698,"type":150,"component":699,"componentVersion":5,"config":700,"objective":704,"textAlternative":705},"lab-spot-both","line-spotter",{"modes":701,"rounds":677},[702,703],"kinds","pairs","Name lines, rays and segments, and classify pairs as parallel, perpendicular or intersecting, in one mixed game.","This round mixes both kinds of question.\n\n**Kinds rounds** show one straight figure through two labelled points and offer four names, such as line AB, segment AB, ray AB and ray BA. Decide from its ends: no arrows means **segment**; arrows at both ends means **line**; one arrow means **ray**, named by the point it starts from.\n\n**Pairs rounds** show two lines. If the perpendicular gap is the same everywhere and they will never meet, they are **parallel**. If they meet at a right angle (the small-square corner), they are **perpendicular**. If they meet at any other angle, or would meet if extended, they are **intersecting**.\n\nRemember that perpendicular lines need not be horizontal and vertical: the pairs are turned to random angles. A small square in the corner, when shown, marks a right angle.",{"id":707,"type":395,"title":708,"items":709},"steps-draw-parallel","Drawing a parallel line with a ruler and set square",[710,714,718,722],{"title":711,"tag":712,"text":713},"Place the set square","edge on line l","Lay one edge of the set square exactly along the given line l.",{"title":715,"tag":716,"text":717},"Add the ruler","against another edge","Hold a ruler firmly against a second edge of the set square.",{"title":719,"tag":720,"text":721},"Slide","keep the ruler still","Slide the set square along the ruler until its first edge reaches point P.",{"title":723,"tag":724,"text":725},"Draw","through P","Draw along that edge. The new line passes through P and is parallel to l, because the edge kept the same direction.",{"id":727,"type":395,"title":728,"items":729},"steps-draw-perp","Drawing a perpendicular through a point with a set square",[730,734,738],{"title":731,"tag":732,"text":733},"Line up","one short edge on l","Place one of the two edges that form the set square's right angle along line l.",{"title":735,"tag":736,"text":737},"Slide to P","along l","Slide it along l until the other right-angle edge passes through point P.",{"title":723,"tag":739,"text":740},"along the upright edge","Draw along that edge. The new line meets l at 90° and passes through P.",{"id":742,"type":47,"variant":79,"title":743,"markdown":744},"hv-note","Horizontal and vertical","A **horizontal** line is level, like the surface of still water. A **vertical** line points straight up and down, like a plumb line. Horizontal and vertical lines are always perpendicular. On a page we usually call left-to-right \"horizontal\" and top-to-bottom \"vertical\", even though the page itself may be lying flat on a desk.",{"id":746,"type":53,"title":747,"eyebrow":748,"navLabel":749},"ch11","Lines in plans, wiring and maps","Chapter 11","11 Plans and maps",{"id":751,"type":43,"markdown":752},"plans-intro","The precise language of this lesson is used every day by people who draw plans. An architect's floor plan, an electrician's wiring layout and a map all rely on the same few ideas: segments with exact lengths, lines that are parallel or perpendicular, and points named so that nobody can be confused.",{"id":754,"type":135,"title":755,"problem":756,"steps":757},"we-wiring","Reading a wiring plan","An electrician's plan shows a switch board at point S on a wall, 1.2 m above the floor. A wire runs vertically up from S to point T at the ceiling, 3 m above the floor, then horizontally along the wall to a light at point L, 2.5 m from T. How long is the wire? Name the segments and say how they are related.",[758,759,760,761,762],"Segment ST is vertical. Its length is 3 − 1.2 = 1.8 m.","Segment TL is horizontal, along the top of the wall, 2.5 m long.","Total wire = ST + TL = 1.8 + 2.5 = **4.3 m**.","ST is vertical and TL is horizontal, so ST ⊥ TL: they meet at a right angle at T.","The shortest path from S to L would be the slanting segment SL, but wires follow vertical and horizontal lines so that people can guess where they are.",{"id":764,"type":43,"markdown":765},"map-grid","A **map grid** is a set of evenly spaced parallel lines running north–south, crossed at right angles by another set running east–west. Every square has a name made from its column and row, like a seat in a cinema. A place is found by reading its column first and then its row. Official topographic sheets, such as those of the **Survey of India** used for planning roads and railways, are printed with a national grid like this.\n\nOld Jaipur, planned in 1727 under Sawai Jai Singh II, was laid out with wide straight roads crossing at right angles into large blocks, so even the city itself is a grid of perpendicular lines.",{"id":767,"type":95,"caption":768,"columns":769,"rows":773},"table-grid-ref","A small map grid (columns A to D, rows 1 to 3)",[770,771,772],"Place","Grid square","What the lines do",[774,778,782,786],[775,776,777],"School","B2","Between the 2nd and 3rd north–south lines, and between the 2nd and 3rd east–west lines",[779,780,781],"Railway station","D1","In the last column, first row",[783,784,785],"Temple","A3","First column, last row",[787,788,789],"Kirana shop","B3","Directly below the school: same column, next row",{"id":791,"type":250,"itemId":792,"prompt":793,"check":794,"hints":796,"feedback":799},"prac-grid-squares","lines.understand-grid-squares","A map grid has 6 north–south lines and 5 east–west lines, all evenly spaced, with the outer lines forming the border. How many small squares does it have?",{"kind":324,"answer":795,"tolerance":326},20,[797,798],"6 lines make 5 gaps across.","5 lines make 4 gaps down.",{"correct":800,"incorrect":801},"Yes: 5 × 4 = 20 squares.","6 north–south lines leave 5 columns; 5 east–west lines leave 4 rows. Squares = 5 × 4 = **20**.",{"id":803,"type":47,"variant":273,"title":804,"markdown":805},"mis-student-answers","Three mistakes teachers see every year","1. **Writing \\\"line AB = 5 cm\\\".** Lines have no length. Write AB = 5 cm or segment AB.\n2. **Naming a ray from the wrong end.** A ray that starts at Q and passes through P is ray **QP**, not ray PQ. Point at the start first, then at the direction.\n3. **Saying \\\"perpendicular means vertical\\\".** A slanted roof beam can be perpendicular to another slanted beam. Perpendicular describes how **two** lines meet (at 90°), not how one line points.",{"id":807,"type":135,"title":808,"problem":809,"steps":810},"we-naming-four","Naming problem: four points on a line","Points A, B, C and D lie on a line in that order. (1) How many different segments are there? (2) Which rays are the same as ray BC? (3) Name a pair of opposite rays with end point C.",[811,812,813],"(1) Pairs of points: AB, AC, AD, BC, BD, CD. That is **6** segments (4 × 3 ÷ 2).","(2) Ray BC starts at B and heads towards C and D. So **ray BD** is the same ray. (Ray BA is not: it heads the other way.)","(3) From C, one ray heads back through B and A (ray CB, the same as ray CA), and one heads on through D (ray CD). **Ray CB and ray CD** are opposite rays.",{"id":815,"type":250,"itemId":816,"prompt":817,"check":818,"hints":829,"feedback":832},"prac-opposite","lines.understand-opposite","P, Q, R and S lie on a line in that order. Which pair are opposite rays?",{"kind":254,"options":819,"correct":828},[820,822,824,826],{"id":202,"label":821},"Ray QP and ray QR",{"id":205,"label":823},"Ray PQ and ray RS",{"id":208,"label":825},"Ray QR and ray QS",{"id":263,"label":827},"Ray PS and ray SP",[202],[830,831],"Opposite rays share the same end point.","They must point in opposite directions.",{"correct":833,"incorrect":834},"Yes: both start at Q and point opposite ways.","Opposite rays share an end point and point opposite ways: **ray QP and ray QR**. Ray QR and ray QS are the same ray; ray PS and ray SP start at different points.",{"id":836,"type":135,"title":837,"problem":838,"steps":839},"we-divider-vs-ruler","Ruler or divider for a tiny segment?","Segment MN in a textbook figure is very short. Arjun lays his thick ruler on it and reads 1.6 cm from the side. Then he opens a divider on M and N, moves it to the ruler and, looking from directly above, reads the points at 3.0 cm and 4.4 cm. Which result should he trust, and what is MN?",[840,841,842,843],"With the divider, MN = 4.4 − 3.0 = **1.4 cm**.","The divider's points touch the paper at M and N, so there is no gap for parallax, and the reading was taken from directly above.","The ruler reading was taken from the side through thick plastic, so it may be off by a millimetre or two: 1.6 − 1.4 = 0.2 cm = 2 mm too long.","Trust the divider: **MN = 1.4 cm**. For short segments a small error is a large fraction of the length.",{"id":845,"type":53,"title":846,"eyebrow":847,"navLabel":848},"ch12","Mix-ups, practice and summary","Chapter 12","12 Wrap-up",{"id":850,"type":135,"title":851,"problem":852,"steps":853},"we-parallax-size","How big is a parallax error?","The true length of a segment is 6.1 cm. Reading from the side, Meena gets 6.3 cm; reading from directly above, she gets 6.1 cm. By how many millimetres was her side reading wrong, and was it too big or too small?",[854,855,856],"Error = 6.3 − 6.1 = 0.2 cm.","0.2 cm × 10 = **2 mm**, too big.","2 mm sounds tiny, but on a 6 cm segment it is about 1 part in 30. When segments are added together for a construction, such errors pile up.",{"id":858,"type":95,"caption":859,"columns":860,"rows":864},"table-mixups","Common mix-ups and how to fix them",[861,862,863],"Mix-up","What is true","Quick check",[865,869,873,877,881,885,889],[866,867,868],"Line AB is 6 cm long","Lines have no length; segments do","Does it have two end points?",[870,871,872],"Ray AB = ray BA","The first letter is the start; they point opposite ways","Where does it start?",[874,875,876],"They don't touch here, so they're parallel","Extend them; measure the gap at two places","Is the gap the same everywhere?",[878,879,880],"Perpendicular means one line is upright","Any two lines meeting at 90°","Fit a folded-paper corner",[882,883,884],"Three points always make a triangle","Only if they are non-collinear","Lay a ruler through two of them",[886,887,888],"Concurrent means intersecting","Concurrent means three or more lines through one point","How many lines share the point?",[890,891,892],"Measuring from the ruler's end","Start at the 0 mark, or subtract","Is the 0 on the start point?",{"id":894,"type":150,"component":458,"componentVersion":5,"config":895,"objective":950,"textAlternative":951},"lab-sort-props",{"prompt":896,"bins":897,"items":910,"seconds":326},"Which of the three does each statement describe? Choose the most exact bin.",[898,901,904,907],{"id":899,"label":900},"segment","Segment only",{"id":902,"label":903},"ray","Ray only",{"id":905,"label":906},"line","Line only",{"id":908,"label":909},"all","All three",[911,915,919,923,927,931,935,939,943,946],{"id":912,"label":913,"bin":899,"why":914},"two-ends","Has exactly two end points","Only a segment stops at both ends.",{"id":916,"label":917,"bin":902,"why":918},"one-end","Has exactly one end point","A ray starts at one point and never stops.",{"id":920,"label":921,"bin":905,"why":922},"no-end","Has no end points","A line goes on forever both ways.",{"id":924,"label":925,"bin":899,"why":926},"length","Has a length you can measure","Only something with two ends has a length.",{"id":928,"label":929,"bin":908,"why":930},"straight","Is perfectly straight","Lines, rays and segments are all straight.",{"id":932,"label":933,"bin":902,"why":934},"order","Its name changes meaning if you swap the two letters","Ray AB starts at A; ray BA starts at B.",{"id":936,"label":937,"bin":905,"why":938},"both-ways","Goes on forever in both directions","Only a line has no end at either side.",{"id":940,"label":941,"bin":908,"why":942},"infinite-points","Contains endlessly many points","Even a tiny segment has endlessly many points.",{"id":505,"label":944,"bin":899,"why":945},"Has a midpoint","A midpoint needs two ends to be halfway between.",{"id":947,"label":948,"bin":908,"why":949},"in-plane","Can be drawn on a flat page (as a picture)","All three are drawn on a plane, with arrows standing for \"keeps going\".","Decide which statements describe only segments, only rays, only lines, or all three.","Ten statements, four bins.\n\n**Segment only:** has exactly two end points; has a length you can measure; has a midpoint.\n\n**Ray only:** has exactly one end point; its name changes meaning if you swap the letters (ray AB starts at A, ray BA at B).\n\n**Line only:** has no end points; goes on forever in both directions.\n\n**All three:** perfectly straight; contain endlessly many points; can be drawn on a flat page as a picture.",{"id":953,"type":954,"title":955,"terms":956},"glossary-understand","glossary","Vocabulary for lines, precisely",[957,961,964,968,971,975,978,982,986,990,993,997,1001,1005,1009,1013],{"term":958,"meaning":959,"example":960},"Undefined term","A starting idea that is described but not defined, because defining it would need other words. Point, line and plane are the three undefined terms of school geometry.","A point is described as a position with no size",{"term":451,"meaning":962,"example":963},"Points that lie on one straight line.","Three beads on a tight thread",{"term":965,"meaning":966,"example":967},"Non-collinear points","Points that do not all lie on one line.","The corners of a triangle",{"term":551,"meaning":969,"example":970},"Three or more lines passing through one point.","Spokes of a wheel, extended",{"term":972,"meaning":973,"example":974},"Point of concurrence","The common point of concurrent lines.","The hub of the wheel",{"term":221,"meaning":976,"example":977},"Two rays with the same end point pointing in opposite directions; together they form a line.","Ray QP and ray QR, with Q between P and R",{"term":979,"meaning":980,"example":981},"Midpoint","The point that divides a segment into two equal parts.","M with AM = MB",{"term":983,"meaning":984,"example":985},"Bisect","To cut into two equal parts.","The fold bisects the segment",{"term":987,"meaning":988,"example":989},"Perpendicular bisector","The line through the midpoint of a segment at right angles to it. Every point on it is equally far from both ends.","The crease when you fold A onto B",{"term":637,"meaning":991,"example":992},"A line that crosses two or more lines at different points.","A road crossing two railway rails",{"term":994,"meaning":995,"example":996},"Distance between parallel lines","The length of a segment perpendicular to both lines; it is the same everywhere.","The gap between notebook lines",{"term":998,"meaning":999,"example":1000},"Parallax error","A reading error caused by looking at a scale from the side instead of from directly above.","Reading 5.2 cm instead of 5.0 cm",{"term":1002,"meaning":1003,"example":1004},"Divider","A two-pointed instrument used to transfer a length from a drawing to a ruler.","Measuring a tiny segment accurately",{"term":1006,"meaning":1007,"example":1008},"Skew lines","Lines in space that never meet and are not parallel, because they do not lie in one plane.","A wall-ceiling edge and a floor edge on another wall",{"term":1010,"meaning":1011,"example":1012},"Horizontal","Level, like still water.","The horizon at sea",{"term":1014,"meaning":1015,"example":1016},"Vertical","Straight up and down, like a plumb line.","A flag pole",{"id":1018,"type":1019,"title":1020,"questions":1021},"quiz-understand","quiz","Precise ideas about lines",[1022,1032,1044,1053,1066,1076,1089,1102,1115,1128,1141],{"itemId":1023,"prompt":1024,"options":1025,"correct":263,"why":1031},"lines.understand-q-undefined","Which of these is **not** one of the undefined starting ideas of geometry?",[1026,1027,1028,1029],{"id":202,"label":66},{"id":205,"label":70},{"id":208,"label":74},{"id":263,"label":1030},"Line segment","A segment is defined using points and a line: the part of a line between two end points.",{"itemId":1033,"prompt":1034,"options":1035,"correct":205,"why":1043},"lines.understand-q-two-points","How many lines pass through two different points?",[1036,1037,1039,1041],{"id":202,"label":236},{"id":205,"label":1038},"Exactly one",{"id":208,"label":1040},"Two",{"id":263,"label":1042},"Endlessly many","Two points fix exactly one straight line.",{"itemId":1045,"prompt":1046,"options":1047,"correct":208,"why":1052},"lines.understand-q-one-point","How many lines pass through one point?",[1048,1049,1050,1051],{"id":202,"label":1038},{"id":205,"label":1040},{"id":208,"label":1042},{"id":263,"label":236},"Every direction gives a different line through the point.",{"itemId":1054,"prompt":1055,"options":1056,"correct":202,"why":1065},"lines.understand-q-ray-name","Ray PQ starts at…",[1057,1059,1061,1063],{"id":202,"label":1058},"P",{"id":205,"label":1060},"Q",{"id":208,"label":1062},"Either end",{"id":263,"label":1064},"The midpoint of PQ","The first letter of a ray's name is always its end point.",{"itemId":1067,"prompt":1068,"options":1069,"correct":208,"why":1075},"lines.understand-q-noncollinear","Three non-collinear points. How many different lines pass through pairs of them?",[1070,1071,1072,1073],{"id":202,"label":587},{"id":205,"label":591},{"id":208,"label":595},{"id":263,"label":1074},"6","Lines AB, BC and CA are all different, like the sides of a triangle.",{"itemId":1077,"prompt":1078,"options":1079,"correct":205,"why":1088},"lines.understand-q-concurrent","Concurrent lines are…",[1080,1082,1084,1086],{"id":202,"label":1081},"Two lines that never meet",{"id":205,"label":1083},"Three or more lines through one point",{"id":208,"label":1085},"Lines at right angles",{"id":263,"label":1087},"Lines on different planes","Concurrent means running together: three or more lines through a single common point.",{"itemId":1090,"prompt":1091,"options":1092,"correct":205,"why":1101},"lines.understand-q-parallel-def","Which condition is part of the definition of parallel lines?",[1093,1095,1097,1099],{"id":202,"label":1094},"They are both horizontal",{"id":205,"label":1096},"They lie in the same plane",{"id":208,"label":1098},"They are the same length",{"id":263,"label":1100},"They are 1 cm apart","Lines in different planes can fail to meet without being parallel; those are skew lines.",{"itemId":1103,"prompt":1104,"options":1105,"correct":205,"why":1114},"lines.understand-q-between","B lies between A and C. AB = 2.5 cm and BC = 4.3 cm. AC = ?",[1106,1108,1110,1112],{"id":202,"label":1107},"1.8 cm",{"id":205,"label":1109},"6.8 cm",{"id":208,"label":1111},"6.3 cm",{"id":263,"label":1113},"Cannot tell","AC = AB + BC = 2.5 + 4.3 = 6.8 cm.",{"itemId":1116,"prompt":1117,"options":1118,"correct":205,"why":1127},"lines.understand-q-parallax","To avoid parallax error when reading a ruler you should…",[1119,1121,1123,1125],{"id":202,"label":1120},"Look from the side",{"id":205,"label":1122},"Put your eye directly above the mark",{"id":208,"label":1124},"Measure from the ruler's end",{"id":263,"label":1126},"Use a longer ruler","Looking straight down lines up the mark with the point on the paper.",{"itemId":1129,"prompt":1130,"options":1131,"correct":208,"why":1140},"lines.understand-q-perp-bisector","The perpendicular bisector of AB passes through…",[1132,1134,1136,1138],{"id":202,"label":1133},"A",{"id":205,"label":1135},"B",{"id":208,"label":1137},"The midpoint of AB",{"id":263,"label":1139},"No point of AB","It bisects AB, so it goes through the midpoint, and it meets AB at 90°.",{"itemId":1142,"prompt":1143,"options":1144,"correct":202,"why":1152},"lines.understand-q-transversal","A line that crosses two other lines at two different points is called a…",[1145,1146,1148,1150],{"id":202,"label":637},{"id":205,"label":1147},"Ray",{"id":208,"label":1149},"Bisector",{"id":263,"label":1151},"Diagonal","That is the definition of a transversal.",{"id":1154,"type":1155,"prompt":1156},"reflect-understand","reflection","Explain to a younger friend, without using the word \"infinite\", why a line cannot have a length but a line segment can. Then explain why ray AB and ray BA are different.",{"id":1158,"type":1159,"title":1160,"points":1161},"cheat-understand","summary","Cheat sheet",[1162,1163,1164,1165,1166,1167,1168,1169,1170,1171,1172],"**Undefined terms:** point (position, no size), line (straight, endless both ways), plane (flat, endless). Everything else is defined from these.","**Names:** points by capital letters; line AB or line l; segment AB = segment BA; ray AB starts at A, so ray AB ≠ ray BA.","**Length:** AB with nothing over it is a number: AB = 5 cm. Only segments have length.","**Two points fix one line.** Through one point: endlessly many lines.","**Betweenness:** B on segment AC ⇒ AB + BC = AC. Otherwise AB + BC > AC.","**Measuring:** start at 0 (or subtract readings); eye directly above to avoid parallax; a divider transfers lengths accurately.","**Collinear:** on one line. Three points give 1 line (collinear) or 3 lines (non-collinear).","**Intersecting:** two lines meet in at most one point. **Concurrent:** three or more lines through one point.","**Parallel (∥):** same plane, never meet, constant perpendicular gap. Different planes and never meeting: **skew**.","**Perpendicular (⊥):** meet at 90°; all four angles are right angles. **Perpendicular bisector:** through the midpoint at 90°; its points are equidistant from both ends.","**Transversal:** crosses two or more lines at different points.",{"id":1174,"type":641,"conceptId":1175,"relation":643,"explanation":1176},"conn-shape-u","shape-and-space","Sides of polygons are segments; parallel and perpendicular sides define rectangles, squares and parallelograms.",{"id":1178,"type":641,"conceptId":1179,"relation":686,"explanation":1180},"conn-data","data-handling","Bar graphs and axes rely on perpendicular axes and parallel, equally spaced grid lines.",{"id":1182,"type":1183,"sourceIds":1184},"sources-understand","sources",[1185,1186,1187,1188,1189,1190,1191,1192],"lines-ncert-math-6","lines-ncert-ganita-prakash-6","lines-ncert-math-7","lines-mathsisfun-line","lines-mathsisfun-parallel-perpendicular","lines-britannica-euclidean-geometry","lines-wiki-jaipur","lines-wiki-topographic-map",[1185,1186,1187,1188,1189,1190,1191,1192],"needs_review",{"generatedBy":1196,"notes":1197},"claude-code","Draft generated by a scripted generator; every count and length was computed in Python. Pending owner review.","3c9a4499cc2334ad430482cb75413647e48a8362e6b725720615fe7ff9135c6f",{"component:match-pairs@1":1200,"logic:practice":1201,"component:sort-game@1":1202,"component:line-spotter@1":1203,"source:lines-britannica-euclidean-geometry":1204,"source:lines-mathsisfun-line":1205,"source:lines-mathsisfun-parallel-perpendicular":1206,"source:lines-ncert-ganita-prakash-6":1207,"source:lines-ncert-math-6":1208,"source:lines-ncert-math-7":1209,"source:lines-wiki-jaipur":1210,"source:lines-wiki-topographic-map":1211},"2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","1279a4e23634aba0fd1cb2bb76638a44d6b592131c326e194fdd1bf593a12f93","40d3ed4e883bb30133b96e69ed296da3721d5ef497178eea1c88023a8c2403c9","982f31ecb1e1e17e74ba13872e9e63e218010de6c5653901ebd0db8a12413bc3","ee8cac463d4577f5a05460d731e8e92e3d4268172d2221efc74a4e61e4376345","b7ec4502e5c7912ecccfeb371ab30a19043c5598016c6d51acbc678290fc40a2","219993d3ebc010eeac3a16481eda537da1557ef707d475cbca1ba05a49dfff80","f380f754917bdc6d17093f871ff43201664a75572d28b860df518fdbf8f740be","035dc87f090626f361ac7a1fe771c647be6d32cf29bb0b110a6bf346deae287f","0c0a3178d6df74468b152130b4f03cb06ce82923df444d678aa13d5de0666814",{"state":1213,"reviewer":1214,"selfReview":1215,"reviewedAt":1216,"method":1217},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598307]