[{"data":1,"prerenderedAt":1177},["ShallowReactive",2],{"layer:lines:extend":3},{"layer":4,"contentHash":1149,"dependencyHashes":1150,"approval":1170,"releaseId":1176},{"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":1144,"reviewStatus":1145,"authoring":1146},1,"lines","en","extend","Lines in the wider world","Perspective, skew lines, maps, sport, careers, puzzles and open questions","See parallel lines meet in perspective drawings, find skew lines in rooms and solids, read lines on maps and sports grounds, meet people who use lines at work, and tackle puzzles from pizza cuts to string art.",[13,14,15,16,17],"Explain vanishing points and how projective geometry lets parallel lines meet at infinity.","Identify and count parallel, intersecting and skew edges of solids and rooms.","Describe how latitude and longitude behave as lines on a round Earth, and compute IST from India's standard meridian.","Use parallel and perpendicular lines to describe cricket and badminton markings and real jobs.","Solve counting puzzles with straight lines and explain why string art makes curves.",50,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 50 minutes",{"label":29,"value":30},"Prior knowledge","Understand, Investigate, Deepen",{"label":32,"value":33},"Chapters","12",{"label":35,"value":36},"Labs","Solids, room sort, world match, line spotter",{"label":38,"value":39},"Projects","Five week-long projects",[41,45,51,57,60,85,90,93,98,103,106,111,138,148,164,220,225,230,233,254,259,268,273,278,281,305,333,348,372,384,388,393,455,460,465,468,472,475,481,486,489,494,504,516,520,529,541,546,549,558,567,576,588,592,597,600,604,608,611,635,639,644,668,672,677,680,740,749,761,773,777,786,846,850,876,881,913,923,970,1095,1099,1114,1118,1123,1126],{"id":42,"type":43,"markdown":44},"intro-extend","prose","Lines are not only for geometry homework. Artists bend the rules of parallel lines to make flat pictures look deep. Engineers keep railway rails exactly parallel across thousands of kilometres. Map makers draw lines on a round Earth. Computer games fire millions of rays every second to draw each frame. And some puzzles about straight lines have kept mathematicians busy for centuries.\n\nThis layer takes the ideas from the first four depths into the wider world: three dimensions, maps, art, sport, careers, puzzles and a few questions that nobody has fully answered.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how-e","callout","observation","How to use this layer","Pick the chapters that interest you most; they do not depend strongly on each other. Several chapters end with a **project** you can do over a week. The puzzles get harder towards the end, so don't be discouraged if the last ones take a few days.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","Where parallel lines meet: perspective","Chapter 01","1 Perspective",{"id":58,"type":43,"markdown":59},"perspective","Stand in the middle of a long, straight railway platform and look along the track. The rails are parallel: 1,676 mm apart on India's broad gauge, all the way. Yet they appear to **close in** and meet at a single point on the horizon. The sleepers look shorter and shorter, and closer together.\n\nThis is **perspective**. Things further away make a smaller image in your eye, so a fixed gap looks smaller the further off it is. Any set of parallel lines running away from you seems to meet at one point, called the **vanishing point**, which lies on the **horizon line** at the height of your eyes.",{"id":61,"type":62,"title":63,"items":64},"steps-one-point","steps","Draw a railway track in one-point perspective",[65,69,73,77,81],{"title":66,"tag":67,"text":68},"Horizon","eye level","Draw a horizontal line across your page. This is the horizon, at the height of your eyes.",{"title":70,"tag":71,"text":72},"Vanishing point","one dot","Mark a point V in the middle of the horizon.",{"title":74,"tag":75,"text":76},"Rails","towards V","From two points on the bottom edge of the page, draw straight lines to V. These are the rails.",{"title":78,"tag":79,"text":80},"Sleepers","horizontal","Draw horizontal segments between the rails, getting closer together as they approach V.",{"title":82,"tag":83,"text":84},"Extras","poles and platform","Draw the platform edge and the tops of electric poles as lines also heading to V.",{"id":86,"type":47,"variant":87,"title":88,"markdown":89},"aha-perspective","aha","Parallel on the ground, meeting on the paper","In the real scene the rails are parallel and never meet. In your drawing they are **not parallel**: they intersect at V. A drawing turns parallel lines that run away from you into lines that meet. Lines that run **across** your view (like the sleepers) stay parallel in the drawing. The Florentine architect Filippo Brunelleschi is usually credited with demonstrating the rules of perspective around 1415, and painters of the Renaissance used them to make flat walls look like windows.",{"id":91,"type":43,"markdown":92},"projective","Mathematicians took the artist's trick seriously. In the 1600s Girard Desargues, and later Jean-Victor Poncelet in 1822, developed **projective geometry**. It adds one extra \"point at infinity\" for every direction, where all lines of that direction meet, and a \"line at infinity\" made of all such points: the horizon of the whole plane.\n\nIn projective geometry there are **no parallel lines**: every two different lines meet in exactly one point, either an ordinary one or one at infinity. It sounds like cheating, but it makes many theorems simpler, and it is the geometry that cameras, computer graphics and virtual-reality headsets use every day.",{"id":94,"type":47,"variant":95,"title":96,"markdown":97},"model-perspective","model_limit","What this changes, and what it doesn't","Saying parallel lines \"meet at infinity\" is a **new model** with an extra rule, not a correction of ordinary geometry. On a flat page, with ordinary points only, parallel lines still never meet. Always say which model you are using.",{"id":99,"type":53,"title":100,"eyebrow":101,"navLabel":102},"ch02","Lines in three dimensions: skew lines","Chapter 02","2 Skew lines",{"id":104,"type":43,"markdown":105},"skew","On a page, two lines either meet or are parallel. In a room there is a third possibility. Look at the edge where the front wall meets the floor, running left to right. Now look at the edge where the left wall meets the ceiling, running from front to back. They never meet. But they are not parallel either: they point in different directions. Such lines are called **skew lines**.\n\nSkew lines can only exist in three dimensions, because two lines that lie in one plane must either meet or be parallel. The edges of every box, cupboard and building give plenty of examples.",{"id":107,"type":47,"variant":108,"title":109,"markdown":110},"def-skew","definition","Skew lines","Two lines are **skew** if they do not meet and are not parallel. Skew lines never lie in the same plane. Two lines in space are always exactly one of: **intersecting**, **parallel** or **skew**.",{"id":112,"type":113,"caption":114,"columns":115,"rows":122},"table-solids","table","Pairs of edges of some solids (computed by checking every pair)",[116,117,118,119,120,121],"Solid","Edges","Pairs of edges","Parallel","Intersecting (share a corner)","Skew",[123,128,133],[124,33,125,126,127,127],"Cube or cuboid","66","18","24",[129,130,131,132,126,33],"Triangular prism","9","36","6",[134,135,136,137,126,135],"Square pyramid","8","28","2",{"id":139,"type":140,"title":141,"problem":142,"steps":143},"we-cube-edge","worked_example","One edge of a cube and the other eleven","Pick one edge of a cube, say the bottom-front edge. Sort the other 11 edges into parallel, intersecting and skew.",[144,145,146,147],"**Parallel:** the edges pointing the same way (left to right): the top-front, bottom-back and top-back edges. That is **3**.","**Intersecting:** the edges that share one of its two corners. Each corner of a cube has 3 edges, so each corner gives 2 others: 2 + 2 = **4**.","**Skew:** the rest. 11 − 3 − 4 = **4**. For example, the top-left edge running front to back.","Every edge of a cube is the same, so over all 12 edges: parallel pairs = 12 × 3 ÷ 2 = 18, intersecting pairs = 12 × 4 ÷ 2 = 24, skew pairs = 12 × 4 ÷ 2 = 24. Total 66 = 12 × 11 ÷ 2 ✓",{"id":149,"type":150,"component":151,"componentVersion":5,"config":152,"objective":162,"textAlternative":163},"lab-solids","interactive","shape-explorer",{"solids":153,"polygons":158,"modes":159},[154,155,156,157],"cube","cuboid","triangular-prism","square-pyramid",[],[160,161],"explore","count","Rotate a cube, cuboid, triangular prism and square pyramid, count their edges, and find parallel, intersecting and skew edges.","This lab shows four solids you can rotate: a cube, a cuboid, a triangular prism and a square pyramid. Count their faces, edges and corners, then use them to look for relationships between edges.\n\n- **Cube and cuboid:** 12 edges in three groups of 4 parallel edges. Any edge has 3 parallel partners, 4 edges meeting it at its two corners, and 4 skew edges.\n- **Triangular prism:** 9 edges. The 3 long edges are parallel to each other; each edge of the top triangle is parallel to the matching bottom edge. 6 parallel pairs, 18 intersecting pairs, 12 skew pairs.\n- **Square pyramid:** 8 edges. Only the opposite sides of the square base are parallel (2 pairs). The 4 sloping edges all meet at the top. 18 intersecting pairs and 8 skew pairs.\n\nFor each solid, parallel + intersecting + skew pairs add up to the total number of pairs, edges × (edges − 1) ÷ 2.",{"id":165,"type":150,"component":166,"componentVersion":5,"config":167,"objective":218,"textAlternative":219},"lab-sort-room","sort-game",{"prompt":168,"bins":169,"items":176,"seconds":217},"In a rectangular room, is each pair of edges parallel, intersecting or skew?",[170,172,175],{"id":171,"label":119},"parallel",{"id":173,"label":174},"intersecting","Intersecting",{"id":104,"label":121},[177,181,185,189,193,197,201,205,209,213],{"id":178,"label":179,"bin":171,"why":180},"v-v","Front-left vertical corner and back-right vertical corner","Both are vertical.",{"id":182,"label":183,"bin":171,"why":184},"ff-fc","Front wall–floor edge and front wall–ceiling edge","Both run left to right, one above the other.",{"id":186,"label":187,"bin":173,"why":188},"ff-lf","Front wall–floor edge and left wall–floor edge","They meet in the front-left floor corner.",{"id":190,"label":191,"bin":104,"why":192},"ff-lc","Front wall–floor edge and left wall–ceiling edge","Different directions, different heights: they never meet.",{"id":194,"label":195,"bin":104,"why":196},"ff-blv","Front wall–floor edge and back-left vertical corner","One runs across the front at the bottom; the other stands at the back.",{"id":198,"label":199,"bin":173,"why":200},"flv-fc","Front-left vertical corner and front wall–ceiling edge","They meet at the top front-left corner.",{"id":202,"label":203,"bin":171,"why":204},"ff-bc","Front wall–floor edge and back wall–ceiling edge","Both run left to right.",{"id":206,"label":207,"bin":171,"why":208},"lf-rc","Left wall–floor edge and right wall–ceiling edge","Both run front to back.",{"id":210,"label":211,"bin":173,"why":212},"lf-blv","Left wall–floor edge and back-left vertical corner","They meet at the back-left floor corner.",{"id":214,"label":215,"bin":104,"why":216},"fc-blv","Front wall–ceiling edge and back-left vertical corner","They point in different directions and never meet.",0,"Classify pairs of edges of a rectangular room as parallel, intersecting or skew.","Stand in a rectangular room (or imagine one) and check each pair of edges.\n\n**Parallel:** front-left and back-right vertical corners; the front wall's floor edge and ceiling edge; the front wall–floor edge and back wall–ceiling edge; the left wall–floor edge and the right wall–ceiling edge.\n\n**Intersecting:** the front wall–floor edge and the left wall–floor edge (they share a corner); the front-left vertical corner and the front wall–ceiling edge; the left wall–floor edge and the back-left vertical corner.\n\n**Skew:** the front wall–floor edge and the left wall–ceiling edge; the front wall–floor edge and the back-left vertical corner; the front wall–ceiling edge and the back-left vertical corner.\n\nTest: two edges on the same wall, floor or ceiling are never skew.",{"id":221,"type":47,"variant":222,"title":223,"markdown":224},"nuance-planes","nuance","Lines and planes in space","In three dimensions, **two planes** either are parallel (floor and ceiling) or meet in a **line** (two walls meet in a vertical edge). A **line and a plane** either are parallel, meet at exactly one point (a flag pole through the ground), or the line lies in the plane (a floor edge lying on the floor). A line perpendicular to the floor is perpendicular to **every** line on the floor through its foot.",{"id":226,"type":53,"title":227,"eyebrow":228,"navLabel":229},"ch03","Lines on maps and on a round Earth","Chapter 03","3 Maps",{"id":231,"type":43,"markdown":232},"maps","Maps are covered in lines. **Lines of latitude** run east–west and tell you how far north or south you are; **lines of longitude** (meridians) run north–south and tell you how far east or west.\n\nLines of latitude never meet, and stay the same distance apart along the Earth's surface, so they behave like parallel lines: that is why they are also called **parallels**. Lines of longitude are different: they are far apart at the equator, get closer together as you go north or south, and **all meet** at the North and South Poles. Every line of longitude crosses every line of latitude at a right angle.",{"id":234,"type":235,"tone":236,"items":237},"spec-india-lines","spec","blue",[238,242,246,250],{"label":239,"big":240,"value":241},"Standard meridian","82°30′ E","India's time (IST) is set by this line of longitude, which passes near Mirzapur in Uttar Pradesh.",{"label":243,"big":244,"value":245},"Tropic of Cancer","about 23.4° N","Crosses eight states, from Gujarat in the west to Mizoram in the east.",{"label":247,"big":248,"value":249},"IST","UTC + 5:30","82.5° ÷ 15° per hour = 5.5 hours ahead of Greenwich.",{"label":251,"big":252,"value":253},"Equator","0°","The only line of latitude that is a great circle.",{"id":255,"type":47,"variant":256,"title":257,"markdown":258},"mis-map-lines","misconception","“Longitudes are parallel because they look parallel on the map”","On many world maps (such as the Mercator map used in lots of atlases), lines of longitude are drawn as parallel vertical lines. That is a trick of the **projection**, the method of flattening a round Earth onto paper. It stretches the far north and south: Greenland looks as big as Africa, although Africa is about 14 times larger. On a globe, meridians clearly meet at the poles.",{"id":260,"type":140,"title":261,"problem":262,"steps":263},"we-time","Why India is 5 hours 30 minutes ahead","The Earth turns 360° in 24 hours. India's standard meridian is 82.5° east of Greenwich. How far ahead of Greenwich time is Indian Standard Time?",[264,265,266,267],"360° in 24 hours means 360 ÷ 24 = 15° every hour.","82.5° ÷ 15° per hour = 5.5 hours.","5.5 hours = **5 hours 30 minutes**, so IST = UTC + 5:30.","Each line of longitude is a \"time line\": places on the same meridian have the same sun time.",{"id":269,"type":47,"variant":270,"title":271,"markdown":272},"example-flight","example","Why flights to America go over the Arctic","On a flat map, the shortest route from Delhi to San Francisco looks as if it should head east over the Pacific. On a globe, stretch a thread tightly between the two cities and it passes far to the north: the great circle through the two cities climbs to about 75° N, well inside the Arctic Circle. That thread follows a **great circle**, the sphere's version of a straight line. Nonstop flights between India and San Francisco use polar routes for exactly this reason: Air India's first Bengaluru–San Francisco flight, on 10 January 2021, crossed the Arctic Ocean almost over the North Pole. The \"straight line\" on a ball does not look straight on a flat map.",{"id":274,"type":53,"title":275,"eyebrow":276,"navLabel":277},"ch04","Lines by the rulebook: sport","Chapter 04","4 Sport",{"id":279,"type":43,"markdown":280},"sport","Sports grounds are some of the most precisely drawn line diagrams in everyday life. Rules decide which lines are parallel, which are perpendicular and exactly how far apart they are.",{"id":282,"type":113,"caption":283,"columns":284,"rows":288},"table-cricket","The creases at one end of a cricket pitch (MCC Laws)",[285,286,287],"Line","Where","Relationship",[289,293,297,301],[290,291,292],"Bowling crease","Through the centres of the stumps, 2.64 m (8 ft 8 in) long","Parallel to the popping crease",[294,295,296],"Popping crease","1.22 m (4 ft) in front of the bowling crease","Parallel to the bowling crease; perpendicular to the return creases",[298,299,300],"Return creases","Two lines 1.32 m (4 ft 4 in) either side of the middle of the pitch","Perpendicular to the popping and bowling creases; parallel to each other",[302,303,304],"Pitch","20.12 m (22 yards) between the two sets of stumps","The two bowling creases are parallel, one at each end",{"id":306,"type":113,"caption":307,"columns":308,"rows":312},"table-badminton","A badminton court (BWF rules)",[309,310,311],"Measure","Size","Lines involved",[313,317,321,325,329],[314,315,316],"Full length","13.40 m","Between the two back boundary lines, which are parallel",[318,319,320],"Doubles width","6.10 m","Between the outer side lines, which are parallel",[322,323,324],"Singles width","5.18 m","Between the inner side lines",[326,327,328],"Short service line","1.98 m from the net","Parallel to the net",[330,331,332],"Doubles long service line","0.76 m inside the back line","Parallel to the back line",{"id":334,"type":335,"itemId":336,"prompt":337,"check":338,"hints":343,"feedback":345},"prac-badminton","practice","lines.extend-badminton-gap","On a badminton court the doubles width is 6.10 m and the singles width is 5.18 m, with the singles court centred. How far apart, in metres, are each outer side line and the inner side line next to it?",{"kind":339,"answer":340,"tolerance":341,"unit":342},"number",0.46,0.001,"m",[344],"The extra width is shared equally between the two sides.",{"correct":346,"incorrect":347},"Yes: (6.10 − 5.18) ÷ 2 = 0.46 m.","Extra width = 6.10 − 5.18 = 0.92 m, shared between two sides: 0.92 ÷ 2 = **0.46 m**. Those two lines are parallel and 0.46 m apart.",{"id":349,"type":113,"caption":350,"columns":351,"rows":355},"table-gauges","Railway gauges: the fixed distance between parallel rails",[352,353,354],"Gauge","Distance between rails","Where you meet it",[356,360,364,368],[357,358,359],"Broad gauge","1,676 mm","Most of Indian Railways' main lines",[361,362,363],"Standard gauge","1,435 mm","Many metro lines in Indian cities; most railways in Europe and China",[365,366,367],"Metre gauge","1,000 mm","Some older branch lines and heritage routes",[369,370,371],"Narrow gauge","762 mm or 610 mm","Hill railways such as the Darjeeling Himalayan Railway (610 mm)",{"id":373,"type":335,"itemId":374,"prompt":375,"check":376,"hints":379,"feedback":381},"prac-gauge","lines.extend-gauge-difference","How much wider, in millimetres, is broad gauge (1,676 mm) than standard gauge (1,435 mm)?",{"kind":339,"answer":377,"tolerance":217,"unit":378},241,"mm",[380],"Subtract the two gauges.",{"correct":382,"incorrect":383},"Right: 1,676 − 1,435 = 241 mm.","1,676 − 1,435 = **241 mm**. A train built for one gauge cannot run on the other, because its wheels are fixed to fit one exact distance between parallel rails.",{"id":385,"type":47,"variant":222,"title":386,"markdown":387},"nuance-curved-track","Parallel on a curve?","On a bend, the two rails are curves, not straight lines, so strictly they are not parallel lines. But they stay the **same distance apart** all the way round, like two circles with the same centre. Mathematicians call such curves **parallel curves**. The outer rail on a bend is also raised slightly (this is called cant) so that the train leans into the curve.",{"id":389,"type":53,"title":390,"eyebrow":391,"navLabel":392},"ch05","Who uses lines at work?","Chapter 05","5 Careers",{"id":394,"type":395,"title":396,"prompt":397,"options":398},"explorer-careers","explorer","Lines at work","Pick a job to see how it uses lines.",[399,411,422,433,444],{"id":400,"label":401,"chain":402,"badge":407,"note":410},"surveyor","Surveyor",[403,404,405,406],"Fix two points","Sight a straight line","Measure angles","Draw the map",{"text":408,"tone":409},"Lines on the land","yes","Surveyors fix the exact positions of points on land using instruments that sight perfectly straight lines. The Great Trigonometrical Survey of India, begun in 1802, measured a chain of triangles across the country; it later calculated the height of Mount Everest.",{"id":412,"label":413,"chain":414,"badge":419,"note":421},"architect","Architect",[415,416,417,418],"Plan view","Parallel walls","Perpendicular corners","Perspective drawings",{"text":420,"tone":409},"Lines on paper","Architects draw floor plans with parallel and perpendicular walls, and perspective views to show clients what a building will look like. Chandigarh, planned by Le Corbusier's team in the 1950s, is laid out as a grid of roads meeting at right angles.",{"id":423,"label":424,"chain":425,"badge":430,"note":432},"rail","Railway engineer",[426,427,428,429],"Two rails","Fixed gauge","Sleepers across","Gentle curves",{"text":431,"tone":409},"Parallel for 1000s of km","Railway engineers keep the rails exactly parallel (1,676 mm apart on broad gauge) and lay straight stretches joined by carefully designed curves. Indian Railways runs one of the largest networks in the world.",{"id":434,"label":435,"chain":436,"badge":441,"note":443},"carpenter","Carpenter or mason",[437,438,439,440],"Chalk line","Plumb line","Spirit level","Try square",{"text":442,"tone":409},"Lines by hand","A snapped chalk line marks a straight line, a plumb line gives a vertical, a spirit level gives a horizontal, and a try square checks right angles. Together they make walls vertical and shelves level.",{"id":445,"label":446,"chain":447,"badge":452,"note":454},"graphics","Game designer",[448,449,450,451],"Camera","A ray through each pixel","Hit an object","Colour the pixel",{"text":453,"tone":409},"Millions of rays","Many modern games and animated films use **ray tracing**: the computer sends a ray from the virtual camera through every pixel of the screen and works out what it hits, and where light rays from lamps bounce. A single frame can need millions of rays.",{"id":456,"type":47,"variant":457,"title":458,"markdown":459},"tryit-plumb","try_it","Build your own plumb line and level","Tie a small heavy object, such as a key or a nut, to a thread and let it hang still: that thread is **vertical**. Fill a clear plastic bottle three-quarters with water and lay it on its side: the water surface is **horizontal**. Use them to check whether a door frame is truly vertical and a shelf truly level. Is the door frame perpendicular to the floor?",{"id":461,"type":53,"title":462,"eyebrow":463,"navLabel":464},"ch06","Rays of light, wires and circuits","Chapter 06","6 Light and wires",{"id":466,"type":43,"markdown":467},"light","Physics borrows the word **ray** directly from geometry. Light travels in straight lines, so a thin beam from a torch or the Sun is drawn as a ray: an end point at the source and an arrow showing the direction. That is why shadows have sharp straight edges, why you cannot see round corners, and why a **pinhole camera** works: rays from the top of a tree pass through the tiny hole and land at the bottom of the screen, making an upside-down image.",{"id":469,"type":47,"variant":457,"title":470,"markdown":471},"tryit-pinhole","A pinhole viewer","Cut a square hole in one end of a shoebox and tape foil over it. Poke a small pinhole in the foil. Tape tracing paper over a hole at the other end. Point the pinhole at a bright window from inside a dark room: an upside-down picture of the window appears on the tracing paper. Draw rays from the top and bottom of the window through the pinhole to explain why it is upside down. Never look at the Sun, even through the box.",{"id":473,"type":43,"markdown":474},"wires","Look at a circuit diagram in a science book: every wire is drawn as a straight segment, running either horizontally or vertically and turning at right angles. The real wires may be tangled, but drawing them as parallel and perpendicular segments makes the diagram far easier to read. Overhead power lines are strung parallel to each other between pylons, kept a safe distance apart so that they never touch.",{"id":476,"type":477,"conceptId":478,"relation":479,"explanation":480},"conn-electricity","connection","electricity","applied_in","Circuit diagrams draw wires as horizontal and vertical segments, and overhead power lines are strung parallel and kept apart.",{"id":482,"type":53,"title":483,"eyebrow":484,"navLabel":485},"ch07","Puzzles with straight lines","Chapter 07","7 Puzzles",{"id":487,"type":43,"markdown":488},"puzzles-intro","Straight lines lead to some of the best puzzles in mathematics. Try each one before reading the hints.",{"id":490,"type":47,"variant":491,"title":492,"markdown":493},"q-nine-dots","question","The nine dots","Draw nine dots in a 3 by 3 square. Join all nine with **four straight segments**, drawn without lifting your pencil. (Hint: the famous phrase \"think outside the box\" comes from this puzzle. Your segments are allowed to go beyond the square of dots.)",{"id":495,"type":140,"title":496,"problem":497,"steps":498},"we-caterer","Cutting a pizza with straight cuts","What is the greatest number of pieces you can cut a round pizza into with n straight cuts? (The pieces need not be the same size.)",[499,500,501,502,503],"1 cut: 2 pieces. 2 cuts crossing: 4 pieces. 3 cuts, each crossing both others at different points: 7 pieces.","Why? The k-th cut can cross the earlier k − 1 cuts at most once each, so it passes through at most k regions and splits each in two: it adds at most **k** pieces.","So the maximum is 1 + (1 + 2 + 3 + … + n) = 1 + n × (n + 1) ÷ 2.","n = 4: 1 + 10 = **11**; n = 5: **16**; n = 10: 1 + 55 = **56**.","This is sometimes called the lazy caterer's sequence: 2, 4, 7, 11, 16, 22…",{"id":505,"type":335,"itemId":506,"prompt":507,"check":508,"hints":510,"feedback":513},"prac-caterer","lines.extend-pizza-six","What is the greatest number of pieces a pizza can be cut into with 6 straight cuts?",{"kind":339,"answer":509,"tolerance":217},22,[511,512],"The formula is 1 + n × (n + 1) ÷ 2.","6 × 7 ÷ 2 = 21.",{"correct":514,"incorrect":515},"Yes: 1 + 21 = 22.","1 + 6 × 7 ÷ 2 = 1 + 21 = **22**.",{"id":517,"type":47,"variant":491,"title":518,"markdown":519},"q-orchard","The orchard puzzle","A gardener wants to plant **10 trees** in **5 straight rows** with **4 trees in each row**. That seems impossible: 5 × 4 = 20 trees. The trick is that rows can **share** trees. Hint: draw a five-pointed star. How many points are there where its lines cross or end? This kind of question, called an orchard-planting problem, is still being researched for larger numbers of trees.",{"id":521,"type":140,"title":522,"problem":523,"steps":524},"we-grid-lines","Lines through a 3 by 3 grid of dots","Nine dots form a 3 by 3 square grid. How many different straight lines pass through at least two of the dots?",[525,526,527,528],"If no three dots were collinear there would be 9 × 8 ÷ 2 = 36 lines.","But there are **8** lines containing 3 dots: 3 rows, 3 columns and 2 diagonals.","Each such line was counted 3 times (once for each pair of its 3 dots) instead of once, so subtract 2 for each: 36 − 8 × 2 = 20.","No line contains 4 dots, and every other pair of dots is on its own line. Answer: **20** lines. A computer check of every pair agrees.",{"id":530,"type":335,"itemId":531,"prompt":532,"check":533,"hints":535,"feedback":538},"prac-points-two-lines","lines.extend-triangles-parallel","Four points are marked on line l and three on a line m parallel to l. How many triangles have all three corners among these seven points?",{"kind":339,"answer":534,"tolerance":217},30,[536,537],"Count all groups of 3 from 7 points.","Remove groups that lie on one line: 3 points from l, or all 3 from m.",{"correct":539,"incorrect":540},"Right: 35 − 4 − 1 = 30.","Groups of 3 from 7: 35. Collinear groups: 4 from l (choose 3 of 4) and 1 from m. Triangles = 35 − 4 − 1 = **30**.",{"id":542,"type":53,"title":543,"eyebrow":544,"navLabel":545},"ch08","Olympiad corner","Chapter 08","8 Olympiad corner",{"id":547,"type":43,"markdown":548},"olympiad-intro","These problems are in the style of mathematics olympiads. Each one needs an idea from this topic (pairs, collinearity, parallel lines) used in a new way. Try each for at least ten minutes before reading the solution.",{"id":550,"type":140,"title":551,"problem":552,"steps":553},"we-oly-triangles","Triangles made by lines","Five lines are drawn in a plane, no two parallel and no three through one point. How many triangles have all three sides lying along these lines?",[554,555,556,557],"Any three of the lines, if no two are parallel and they are not concurrent, meet in three different points and so enclose exactly one triangle.","So each group of 3 lines gives one triangle, and different groups give different triangles.","Groups of 3 from 5 lines: 5 × 4 × 3 ÷ 6 = **10** triangles.","If two of the lines were parallel, every group containing both would fail: 3 groups lost, leaving 7.",{"id":559,"type":140,"title":560,"problem":561,"steps":562},"we-oly-circle","Lines and a circle","What is the greatest possible number of intersection points of 4 straight lines and 1 circle?",[563,564,565,566],"Lines with lines: at most one point per pair, 4 × 3 ÷ 2 = 6.","Each line meets a circle in at most 2 points: 4 × 2 = 8.","Total at most 6 + 8 = **14**, reached when all these points are different.","The same idea extends: n lines and 1 circle give at most n(n − 1) ÷ 2 + 2n points.",{"id":568,"type":140,"title":569,"problem":570,"steps":571},"we-oly-diagonals","Where diagonals cross","In a convex octagon, what is the greatest possible number of points **inside** the octagon where two diagonals cross?",[572,573,574,575],"Two diagonals cross inside a convex polygon exactly when their four end points are four different corners, joined 'across'.","Any 4 corners of a convex polygon form a convex quadrilateral, whose two diagonals cross once inside.","So each choice of 4 corners gives one crossing: 8 × 7 × 6 × 5 ÷ 24 = **70**.","In a regular octagon some crossings coincide (several diagonals pass through the centre), so the actual count is smaller; 70 is the maximum.",{"id":577,"type":335,"itemId":578,"prompt":579,"check":580,"hints":582,"feedback":585},"prac-oly-lines-circle","lines.extend-lines-circle","What is the greatest possible number of intersection points of 6 straight lines and 1 circle?",{"kind":339,"answer":581,"tolerance":217},27,[583,584],"Pairs of lines first.","Each line meets the circle at most twice.",{"correct":586,"incorrect":587},"Yes: 15 + 12 = 27.","Line–line: 6 × 5 ÷ 2 = 15. Line–circle: 6 × 2 = 12. Total **27**.",{"id":589,"type":47,"variant":491,"title":590,"markdown":591},"q-grid4","A harder grid","A 3 by 3 grid of dots has 20 lines through at least two dots. A 4 by 4 grid has **62** (a computer check of every pair of dots confirms it). Can you explain the 62 by counting lines with 4, 3 and 2 dots on them? Hint: there are 4 rows, 4 columns and 2 long diagonals with 4 dots each, and 4 shorter diagonals with 3 dots each.",{"id":593,"type":53,"title":594,"eyebrow":595,"navLabel":596},"ch09","Lines in art: from kolam to string art","Chapter 09","9 Art and string art",{"id":598,"type":43,"markdown":599},"art","Artists have played with lines for as long as people have made pictures. **Kolam** and **rangoli** start from a grid of dots, and many designs are built from straight segments between the dots, often with lines of symmetry running through the middle. **Warli** paintings from Maharashtra build people and animals out of triangles, circles and straight lines. Temple walls and jaali screens repeat parallel and perpendicular lines to make patterns that let light through.\n\nIn the 1920s the Dutch painter Piet Mondrian made famous paintings from nothing but black horizontal and vertical lines and blocks of colour: a whole art style built on perpendicular lines. Graphic designers today still use a hidden grid of parallel lines to line up text and pictures on every page and screen.",{"id":601,"type":47,"variant":457,"title":602,"markdown":603},"tryit-kolam","A straight-line kolam","Draw a 5 by 5 grid of dots. Using only straight segments between neighbouring dots (across, down or diagonal), make a design that looks the same when you turn the page upside down. Count how many segments you used, and how many of them are parallel to one another.",{"id":605,"type":477,"conceptId":606,"relation":479,"explanation":607},"conn-patterns-art","patterns","Kolam and rangoli designs repeat straight segments on a dot grid, making shape patterns with symmetry.",{"id":609,"type":43,"markdown":610},"string-art","Here is something that sounds impossible: a **curve** made entirely of **straight segments**. Draw two segments meeting at a corner, like an L. Mark 10 equally spaced points on each, numbered 1 to 10 from the corner outward on one arm and from the far end inward on the other. Join 1 to 1, 2 to 2, and so on.\n\nNo single segment is curved, yet a smooth curve appears where the segments crowd together. The curve (a **parabola**) is **touched** by every one of the segments. Mathematicians call such a curve an **envelope**. The same idea makes the curved look of a suspension bridge's cables and many kolam and rangoli designs drawn from straight strokes.",{"id":612,"type":62,"title":613,"items":614},"steps-string-art","Make a string-art curve",[615,619,623,627,631],{"title":616,"tag":617,"text":618},"Draw the arms","an L or a V","Draw two segments of equal length, 10 cm each, meeting at a point.",{"title":620,"tag":621,"text":622},"Mark points","every 1 cm","Mark 10 points on each arm, 1 cm apart.",{"title":624,"tag":625,"text":626},"Number them","opposite directions","On one arm number 1 to 10 from the corner out; on the other, 1 to 10 from the tip in.",{"title":628,"tag":629,"text":630},"Join matching numbers","10 segments","Join 1–1, 2–2, … 10–10 with a ruler, or with thread through holes in card.",{"title":632,"tag":633,"text":634},"Look","the envelope","A smooth curve appears. Try a V with a sharper angle, or four arms to make a star.",{"id":636,"type":47,"variant":87,"title":637,"markdown":638},"aha-envelope","Straight pieces, curved shape","Every segment is perfectly straight, and no ink is ever curved, yet your eye sees a curve. This is a small version of a big idea in mathematics: a curve can be understood as the limit of many straight pieces. It is how computers draw smooth curves on a screen made of straight rows of pixels.",{"id":640,"type":53,"title":641,"eyebrow":642,"navLabel":643},"ch10","Projects and open questions","Chapter 10","10 Projects",{"id":645,"type":62,"title":646,"items":647},"steps-projects","Projects to try over a week",[648,652,656,660,664],{"title":649,"tag":650,"text":651},"Line map of your street","survey","Draw a map of your street marking every parallel and perpendicular pair you can find: road edges, lamp poles, wires, gates.",{"title":653,"tag":654,"text":655},"Perspective photo","vanishing points","Photograph a long corridor or railway track. Print or trace it and draw lines along the parallel edges. Do they meet at one point?",{"title":657,"tag":658,"text":659},"Room edges census","3D counting","Pick one edge of your room. List every other edge as parallel, intersecting or skew. Does your room behave like a cuboid?",{"title":661,"tag":662,"text":663},"String-art card","envelopes","Make a string-art star with four arms. Photograph the curves that appear.",{"title":665,"tag":666,"text":667},"Shadow line","Katyayana's rule","Use a stick and its shadow's tips in the morning and afternoon to find the east–west line in your garden or terrace.",{"id":669,"type":47,"variant":491,"title":670,"markdown":671},"q-open","Open questions for curious learners","Some questions about straight lines are still open or only partly answered.\n\n- **Is space flat?** Measurements of the early universe suggest that, on the largest scales, space is very nearly flat (Euclidean), but scientists cannot rule out a tiny curvature.\n- **Orchard planting:** for large numbers of trees, what is the largest number of rows of 3 you can make? Exact answers are known only for some cases.\n- **How straight is straight?** Railway engineers join straight track to curves with special transition curves so that trains do not jolt. What shape should the join be?\n- **Nearest-neighbour paths:** what is the shortest network of straight roads joining a set of villages? This \"Steiner tree\" problem becomes very hard for many villages.\n- **Could you tell if you lived on a sphere?** How big would a triangle have to be before you could measure its angles adding to more than 180°?",{"id":673,"type":53,"title":674,"eyebrow":675,"navLabel":676},"ch11","Grids, crossings and courts in Indian life","Chapter 11","11 Grids and courts",{"id":678,"type":43,"markdown":679},"grids-life","Some of the most useful straight lines in daily life are the ones that make **grids**: two families of parallel lines crossing at right angles. Map makers, town planners, sports officials and electricians all rely on them.",{"id":681,"type":395,"title":682,"prompt":683,"options":684},"explorer-grids","Grids and markings you can find","Pick one to see the geometry inside it.",[685,696,707,718,729],{"id":686,"label":687,"chain":688,"badge":693,"note":695},"jaipur","Old Jaipur",[689,690,691,692],"Planned 1727","Straight main roads","Crossing at right angles","Large rectangular blocks",{"text":694,"tone":409},"A city on a grid","Jaipur was planned under Sawai Jai Singh II with broad straight roads meeting at right angles, dividing the walled city into large blocks. Parallel roads make it easy to give directions and to lay drains and water pipes in straight runs.",{"id":697,"label":698,"chain":699,"badge":704,"note":706},"toposheet","Survey of India map",[700,701,702,703],"Grid of parallel lines","Numbered columns and rows","Grid reference","Find any place",{"text":705,"tone":409},"Map grid","Topographic sheets carry evenly spaced grid lines. A grid reference gives the column first and then the row, so a place is fixed by where two perpendicular lines cross, just like a point named by two numbers.",{"id":708,"label":709,"chain":710,"badge":715,"note":717},"kabaddi","Kabaddi court",[711,712,713,714],"13 m × 10 m (men)","Mid line","Baulk lines 3.75 m out","Bonus lines 1 m further",{"text":716,"tone":409},"Parallel crosswise lines","The mid line, baulk lines and bonus lines are all parallel to one another and perpendicular to the side lines. A raider has to cross the baulk line, and touching the bonus line can earn an extra point, so the exact position of each parallel line decides points.",{"id":719,"label":720,"chain":721,"badge":726,"note":728},"zebra","Zebra crossing",[722,723,724,725],"Stripes along the road","Laid side by side","Walk straight across","Shortest path",{"text":727,"tone":409},"Parallel stripes","The long sides of the white stripes run along the road, parallel to the traffic; the walking path crosses perpendicular to the kerb, which is the shortest way across. The high contrast of parallel white stripes makes the crossing easy for drivers to see from far away.",{"id":730,"label":731,"chain":732,"badge":737,"note":739},"wiring","Wiring in walls",[733,734,735,736],"Switch board","Vertical run","Horizontal run near ceiling","Right-angle turns",{"text":738,"tone":409},"Vertical and horizontal","Electricians run concealed wiring in straight vertical and horizontal paths with right-angle turns, so the path is predictable. That uses more wire than a slanting shortcut, but it is much safer when someone later drills into the wall.",{"id":741,"type":140,"title":742,"problem":743,"steps":744},"we-grid-walk","Walking on a street grid","In a grid city, blocks are 200 m long in both directions. Meera walks from a crossing 3 blocks east and 4 blocks north along the streets. How far does she walk? How far would a crow fly in a straight line?",[745,746,747,748],"Along the streets she must walk east and north on perpendicular roads: 3 × 200 + 4 × 200 = 600 + 800 = **1,400 m**.","It does not matter in which order she turns: any route going only east and north has the same length.","The crow flies along the segment joining the start and end. East 600 m and north 800 m are perpendicular, and 600 : 800 : 1,000 is the 3 : 4 : 5 pattern, so the straight line is **1,000 m**.","The straight segment is 400 m shorter, which is why people cut diagonally across open ground when they can.",{"id":750,"type":335,"itemId":751,"prompt":752,"check":753,"hints":755,"feedback":758},"prac-grid-routes","lines.extend-grid-routes","On a street grid, how many different shortest routes are there from a corner to the point 2 blocks east and 2 blocks north, if you may only walk east or north?",{"kind":339,"answer":754,"tolerance":217},6,[756,757],"Every shortest route has 4 steps: 2 east and 2 north.","Choose which 2 of the 4 steps are east.",{"correct":759,"incorrect":760},"Yes: 4 × 3 ÷ 2 = 6 routes.","A shortest route is 4 steps, 2 of them east. Choosing the 2 east steps from 4 gives 4 × 3 ÷ 2 = **6** routes: EENN, ENEN, ENNE, NEEN, NENE, NNEE.",{"id":762,"type":335,"itemId":763,"prompt":764,"check":765,"hints":767,"feedback":770},"prac-wiring-length","lines.extend-wiring-length","Wire runs from a switch 1.3 m above the floor straight up to the ceiling 3.1 m above the floor, then 4.2 m along the wall to a fan point, then 0.5 m straight down. How many metres of wire are used?",{"kind":339,"answer":766,"tolerance":341,"unit":342},6.5,[768,769],"Vertical part up: 3.1 − 1.3.","Then add the horizontal and the last vertical piece.",{"correct":771,"incorrect":772},"Right: 1.8 + 4.2 + 0.5 = 6.5 m.","Up: 3.1 − 1.3 = 1.8 m. Along: 4.2 m. Down: 0.5 m. Total = 1.8 + 4.2 + 0.5 = **6.5 m**.",{"id":774,"type":47,"variant":256,"title":775,"markdown":776},"mis-perp-vertical-e","\\u201cPerpendicular means vertical\\u201d, again","On a staircase handrail, a sloping roof or a tilted map, lines can be perpendicular without either being vertical. On the globe, every line of longitude is perpendicular to the equator, and none of them is \\\"vertical\\\" in any everyday sense. Perpendicular is a relationship between **two** lines: they meet at 90°.",{"id":778,"type":140,"title":779,"problem":780,"steps":781},"we-kabaddi-gaps","Gaps between parallel lines on a kabaddi court","On a men's kabaddi court each baulk line is 3.75 m from the mid line, on either side, and each bonus line is 1 m beyond its baulk line. The court is 13 m long (the size in the current international rules). How far apart are the two baulk lines? The two bonus lines? How far is each bonus line from its end line?",[782,783,784,785],"The two baulk lines are on opposite sides of the mid line: 3.75 + 3.75 = **7.5 m** apart.","Each bonus line is 3.75 + 1 = 4.75 m from the mid line, so the two bonus lines are 4.75 × 2 = **9.5 m** apart.","Each half of the court is 13 ÷ 2 = 6.5 m long, so a bonus line is 6.5 − 4.75 = **1.75 m** from its end line.","Check: 1.75 + 4.75 + 4.75 + 1.75 = 13 m ✓. All these lines are parallel, so each gap is the same everywhere across the court.",{"id":787,"type":150,"component":166,"componentVersion":5,"config":788,"objective":844,"textAlternative":845},"lab-sort-markings",{"prompt":789,"bins":790,"items":795,"seconds":217},"Are these markings parallel or perpendicular to each other?",[791,792],{"id":171,"label":119},{"id":793,"label":794},"perpendicular","Perpendicular",[796,800,804,808,812,816,820,824,828,832,836,840],{"id":797,"label":798,"bin":171,"why":799},"kb-mid-baulk","Kabaddi mid line and a baulk line","Both run across the court, 3.75 m apart.",{"id":801,"label":802,"bin":793,"why":803},"kb-side-mid","Kabaddi side line and mid line","The side line runs along the court; the mid line crosses it at 90°.",{"id":805,"label":806,"bin":171,"why":807},"bd-net-short","Badminton net line and short service line","The short service line is 1.98 m from the net and runs the same way.",{"id":809,"label":810,"bin":793,"why":811},"bd-side-back","Badminton side line and back boundary line","They meet at a square corner of the court.",{"id":813,"label":814,"bin":171,"why":815},"cr-bowling-popping","Bowling crease and popping crease","The popping crease is 1.22 m in front of the bowling crease.",{"id":817,"label":818,"bin":793,"why":819},"cr-return-popping","Return crease and popping crease","Return creases are at right angles to the popping crease.",{"id":821,"label":822,"bin":171,"why":823},"zebra-stripes","Two neighbouring zebra stripes","The stripes are laid side by side along the road.",{"id":825,"label":826,"bin":793,"why":827},"zebra-path-kerb","The walking path over a zebra crossing and the kerb","You cross straight over, the shortest way.",{"id":829,"label":830,"bin":171,"why":831},"rail-rail","The two rails of a straight track","Kept 1,676 mm apart on broad gauge.",{"id":833,"label":834,"bin":793,"why":835},"rail-sleeper","A rail and a sleeper","Sleepers lie across the rails.",{"id":837,"label":838,"bin":171,"why":839},"map-lat","Two east–west lines on a map grid","Grid lines of one family never meet.",{"id":841,"label":842,"bin":793,"why":843},"wire-up-along","A vertical wire run and a horizontal run near the ceiling","Vertical and horizontal lines meet at 90°.","Sort pairs of real markings on courts, roads, tracks, maps and walls into parallel and perpendicular.","Twelve pairs of real markings.\n\n**Parallel:** kabaddi mid line and baulk line; badminton net line and short service line; bowling crease and popping crease; neighbouring zebra stripes; the two rails of a straight track; two east–west grid lines on a map.\n\n**Perpendicular:** kabaddi side line and mid line; badminton side line and back line; return crease and popping crease; the walking path over a zebra crossing and the kerb; a rail and a sleeper; a vertical wire run and a horizontal run near the ceiling.",{"id":847,"type":47,"variant":256,"title":848,"markdown":849},"mis-ray-extend","\\u201cA light ray from a lamp is ray LB whichever way you write it\\u201d","In physics diagrams, arrows on light rays show direction, and the geometry naming rule still applies: a ray that leaves a lamp L and passes a point B is ray **LB**. Ray BL would start at B and head back through the lamp. Students who swap the letters in ray diagrams get reflections and pinhole images the wrong way round.",{"id":851,"type":335,"itemId":852,"prompt":853,"check":854,"hints":870,"feedback":873},"prac-lamp-ray","lines.extend-lamp-ray","Light from a street lamp L passes the top of a pole T and hits the ground at G, with L, T and G collinear. Which ray is the path of the light?",{"kind":855,"options":856,"correct":869},"choice",[857,860,863,866],{"id":858,"label":859},"a","Ray GL",{"id":861,"label":862},"b","Ray LG",{"id":864,"label":865},"c","Ray TL",{"id":867,"label":868},"d","Segment GL only",[861],[871,872],"Where does the light start?","A ray is named from its starting point.",{"correct":874,"incorrect":875},"Yes: the light starts at L and heads through T to G, so it is ray LG (the same as ray LT).","Light starts at the lamp, so the ray begins at L: **ray LG**, which is the same ray as ray LT. Ray GL and ray TL point back towards the lamp.",{"id":877,"type":53,"title":878,"eyebrow":879,"navLabel":880},"ch12","Final challenge and summary","Chapter 12","12 Wrap-up",{"id":882,"type":150,"component":883,"componentVersion":5,"config":884,"objective":911,"textAlternative":912},"lab-match-world","match-pairs",{"prompt":885,"mode":886,"pairs":887},"Match each real-world thing to the kind of lines it shows.","connect",[888,891,894,896,899,902,905,908],{"a":889,"b":890},"Railway rails","Parallel lines",{"a":892,"b":893},"Lines of longitude at the pole","Concurrent lines",{"a":895,"b":109},"A floor edge and an opposite ceiling edge in another direction",{"a":897,"b":898},"A torch beam","A ray",{"a":900,"b":901},"Popping crease and return crease","Perpendicular lines",{"a":903,"b":904},"Rails in a perspective drawing","Lines meeting at a vanishing point",{"a":906,"b":907},"Spokes of a bicycle wheel","Segments meeting at one point",{"a":909,"b":910},"Stumps to stumps on a pitch","A 20.12 m segment","Connect real-world examples to the geometry they show.","Connect each example with its geometry.\n\n- Railway rails: **parallel lines**.\n- Lines of longitude at the North Pole: **concurrent lines** (they all pass through the pole).\n- A floor edge and a ceiling edge running a different direction on another wall: **skew lines**.\n- A torch beam: **a ray**.\n- Popping crease and return crease: **perpendicular lines**.\n- Rails in a perspective drawing: **lines meeting at a vanishing point**.\n- Spokes of a bicycle wheel: **segments meeting at one point** (the hub).\n- Stumps to stumps on a cricket pitch: **a 20.12 m segment**.",{"id":914,"type":150,"component":915,"componentVersion":5,"config":916,"objective":921,"textAlternative":922},"lab-spot-final","line-spotter",{"modes":917,"rounds":920},[918,919],"kinds","pairs",20,"Take the twenty-round final challenge: name every figure and relationship quickly and correctly.","The final challenge: twenty mixed rounds.\n\n**Kinds:** a figure with two end points is a segment; one end point and an arrow is a ray (named from its end point: ray AB or ray BA); arrows at both ends is a line.\n\n**Pairs:** a constant gap that never closes means parallel; a 90° crossing means perpendicular; any other crossing means intersecting.\n\nEverything on the screen is flat, so skew lines cannot appear here; they need three dimensions.",{"id":924,"type":925,"title":926,"terms":927},"glossary-extend","glossary","Vocabulary for the wider world",[928,932,935,939,943,946,950,954,958,962,966],{"term":929,"meaning":930,"example":931},"Perspective","A way of drawing that makes a flat picture look deep, by making distant things smaller.","Railway tracks meeting at the horizon",{"term":70,"meaning":933,"example":934},"The point in a perspective drawing where parallel lines running away from the viewer appear to meet.","Where the rails meet on the horizon",{"term":936,"meaning":937,"example":938},"Horizon line","The line at the viewer's eye level in a perspective drawing.","Where sea meets sky",{"term":940,"meaning":941,"example":942},"Projective geometry","Geometry with extra points at infinity, in which any two lines meet.","Used in cameras and computer graphics",{"term":109,"meaning":944,"example":945},"Lines in space that are neither parallel nor intersecting.","Some pairs of edges of a cuboid",{"term":947,"meaning":948,"example":949},"Latitude","Lines running east–west on the globe; they never meet, so they are also called parallels.","The Tropic of Cancer",{"term":951,"meaning":952,"example":953},"Longitude (meridian)","Lines running north–south on the globe, all meeting at the poles.","82°30′ E, India's standard meridian",{"term":955,"meaning":956,"example":957},"Map projection","A method of flattening the round Earth onto a flat map; every projection distorts something.","The Mercator map",{"term":959,"meaning":960,"example":961},"Ray tracing","A computer graphics method that follows rays of light to colour each pixel.","Realistic reflections in games",{"term":963,"meaning":964,"example":965},"Envelope","A curve touched by every line in a family of lines.","String-art parabola",{"term":967,"meaning":968,"example":969},"Pinhole camera","A dark box where light rays through a small hole form an upside-down image.","Viewing a window on tracing paper",{"id":971,"type":972,"title":9,"questions":973},"quiz-extend","quiz",[974,987,1000,1011,1024,1037,1047,1059,1069,1082],{"itemId":975,"prompt":976,"options":977,"correct":861,"why":986},"lines.extend-q-vanishing","In a one-point perspective drawing, rails that are parallel in real life…",[978,980,982,984],{"id":858,"label":979},"Stay parallel",{"id":861,"label":981},"Meet at the vanishing point",{"id":864,"label":983},"Become perpendicular",{"id":867,"label":985},"Disappear","Parallel lines running away from the viewer are drawn meeting at the vanishing point.",{"itemId":988,"prompt":989,"options":990,"correct":864,"why":999},"lines.extend-q-skew","Which pair of lines can be skew?",[991,993,995,997],{"id":858,"label":992},"Two lines on a page",{"id":861,"label":994},"Two edges of a cuboid on the same face",{"id":864,"label":996},"Two edges of a cuboid on different faces that never meet and point different ways",{"id":867,"label":998},"Two railway rails","Skew lines need three dimensions and never share a plane.",{"itemId":1001,"prompt":1002,"options":1003,"correct":864,"why":1010},"lines.extend-q-cube","How many edges of a cube are skew to a given edge?",[1004,1005,1007,1009],{"id":858,"label":137},{"id":861,"label":1006},"3",{"id":864,"label":1008},"4",{"id":867,"label":135},"11 others: 3 parallel, 4 intersecting, 4 skew.",{"itemId":1012,"prompt":1013,"options":1014,"correct":861,"why":1023},"lines.extend-q-longitude","Lines of longitude all meet at…",[1015,1017,1019,1021],{"id":858,"label":1016},"The equator",{"id":861,"label":1018},"The North and South Poles",{"id":864,"label":1020},"Greenwich",{"id":867,"label":1022},"They never meet","Meridians run north–south and all pass through both poles.",{"itemId":1025,"prompt":1026,"options":1027,"correct":861,"why":1036},"lines.extend-q-ist","IST is UTC + 5:30 because India's standard meridian is…",[1028,1030,1032,1034],{"id":858,"label":1029},"55° E",{"id":861,"label":1031},"82.5° E",{"id":864,"label":1033},"75° E",{"id":867,"label":1035},"90° E","82.5 ÷ 15 = 5.5 hours.",{"itemId":1038,"prompt":1039,"options":1040,"correct":861,"why":1046},"lines.extend-q-popping","The popping crease is ___ to the bowling crease.",[1041,1042,1043,1044],{"id":858,"label":794},{"id":861,"label":119},{"id":864,"label":121},{"id":867,"label":1045},"Concurrent","It is painted parallel to the bowling crease, 1.22 m in front of it.",{"itemId":1048,"prompt":1049,"options":1050,"correct":864,"why":1058},"lines.extend-q-pizza","Most pieces from 4 straight cuts of a pizza?",[1051,1052,1054,1056],{"id":858,"label":135},{"id":861,"label":1053},"10",{"id":864,"label":1055},"11",{"id":867,"label":1057},"16","1 + 4 × 5 ÷ 2 = 11.",{"itemId":1060,"prompt":1061,"options":1062,"correct":864,"why":1068},"lines.extend-q-grid","How many lines pass through at least two dots of a 3 by 3 grid?",[1063,1064,1065,1067],{"id":858,"label":135},{"id":861,"label":1057},{"id":864,"label":1066},"20",{"id":867,"label":131},"36 pairs minus 2 for each of the 8 three-dot lines: 20.",{"itemId":1070,"prompt":1071,"options":1072,"correct":858,"why":1081},"lines.extend-q-projective","In projective geometry, two different lines…",[1073,1075,1077,1079],{"id":858,"label":1074},"Always meet in exactly one point",{"id":861,"label":1076},"Never meet",{"id":864,"label":1078},"Meet twice",{"id":867,"label":1080},"Are always perpendicular","Parallel lines meet at a point at infinity, so every two lines meet once.",{"itemId":1083,"prompt":1084,"options":1085,"correct":861,"why":1094},"lines.extend-q-pinhole","A pinhole camera's image is upside down because…",[1086,1088,1090,1092],{"id":858,"label":1087},"The hole flips colours",{"id":861,"label":1089},"Light travels in straight lines that cross at the hole",{"id":864,"label":1091},"The box is dark",{"id":867,"label":1093},"The paper is thin","Rays from the top go down through the hole and rays from the bottom go up.",{"id":1096,"type":1097,"prompt":1098},"reflect-extend","reflection","Choose one idea from this layer (perspective, skew lines, maps, sport, string art or ray tracing). Explain how it uses the basic ideas of point, line, ray, segment, parallel or perpendicular, and describe a project you could do to explore it further.",{"id":1100,"type":1101,"title":1102,"points":1103},"cheat-extend","summary","Cheat sheet",[1104,1105,1106,1107,1108,1109,1110,1111,1112,1113],"**Perspective:** parallel lines running away from you are drawn meeting at a **vanishing point** on the horizon.","**Projective geometry:** adds points at infinity so any two lines meet exactly once.","**Skew lines:** in 3D, not parallel and not meeting. Two lines in space are intersecting, parallel or skew.","**Cube:** 66 edge pairs = 18 parallel + 24 intersecting + 24 skew. Each edge: 3 parallel, 4 intersecting, 4 skew.","**Maps:** latitudes are parallels; longitudes meet at the poles. IST: 82.5° E ÷ 15° per hour = UTC + 5:30.","**Cricket:** popping crease 1.22 m in front of and parallel to the bowling crease; return creases perpendicular. Pitch 20.12 m.","**Badminton:** 13.40 m by 6.10 m (doubles), 5.18 m wide for singles; short service line 1.98 m from the net.","**Rays in physics:** light travels in straight lines; pinhole cameras and ray tracing depend on it.","**Pizza cuts:** at most 1 + n(n + 1) ÷ 2 pieces with n straight cuts.","**String art:** straight segments can envelope a smooth curve.",{"id":1115,"type":477,"conceptId":1116,"relation":479,"explanation":1117},"conn-shape-e","shape-and-space","Edges of cubes, cuboids, prisms and pyramids show parallel, intersecting and skew lines in three dimensions.",{"id":1119,"type":477,"conceptId":1120,"relation":1121,"explanation":1122},"conn-angles-e","angles","related_to","Lines of longitude meet the equator at right angles; perspective changes the angles we see but not the real ones.",{"id":1124,"type":477,"conceptId":606,"relation":1121,"explanation":1125},"conn-patterns-e","The lazy caterer's sequence 2, 4, 7, 11, 16 grows by 2, 3, 4, 5: a pattern built from straight cuts.",{"id":1127,"type":1128,"sourceIds":1129},"sources-extend","sources",[1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143],"lines-wiki-skew-lines","lines-mcc-laws-creases","lines-mathsisfun-parallel-perpendicular","lines-wiki-parallel-postulate","lines-britannica-euclidean-geometry","lines-ncert-math-7","lines-simpleflying-polar-route","lines-wiki-rail-transport-india","lines-wiki-darjeeling-railway","lines-wiki-kabaddi","lines-wkf-rules-kabaddi","lines-wiki-jaipur","lines-wiki-topographic-map","lines-dani-katyayana-sulvasutra",[1130,1131,1132,1133,1134,1135,1136,1137,1138,1139,1140,1141,1142,1143],"needs_review",{"generatedBy":1147,"notes":1148},"claude-code","Draft generated by a scripted generator; every count and length was computed in Python. Pending owner review.","290da11d5300c58e30c7950b5c9c8b6500f63ae78d10c476acb1adbdd46ce8f6",{"component:shape-explorer@1":1151,"component:sort-game@1":1152,"logic:practice":1153,"component:match-pairs@1":1154,"component:line-spotter@1":1155,"source:lines-britannica-euclidean-geometry":1156,"source:lines-dani-katyayana-sulvasutra":1157,"source:lines-mathsisfun-parallel-perpendicular":1158,"source:lines-mcc-laws-creases":1159,"source:lines-ncert-math-7":1160,"source:lines-simpleflying-polar-route":1161,"source:lines-wiki-darjeeling-railway":1162,"source:lines-wiki-jaipur":1163,"source:lines-wiki-kabaddi":1164,"source:lines-wiki-parallel-postulate":1165,"source:lines-wiki-rail-transport-india":1166,"source:lines-wiki-skew-lines":1167,"source:lines-wiki-topographic-map":1168,"source:lines-wkf-rules-kabaddi":1169},"a8965f19a8548e866e5fcd7f4fec4a9adac35ad54d43c9d5c416cdf3348e6198","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","1279a4e23634aba0fd1cb2bb76638a44d6b592131c326e194fdd1bf593a12f93","40d3ed4e883bb30133b96e69ed296da3721d5ef497178eea1c88023a8c2403c9","705e75261b4a1f9502f10c70908488953d028911f42c2f061fa572d8b5c76388","ee8cac463d4577f5a05460d731e8e92e3d4268172d2221efc74a4e61e4376345","29053ec2c573f86831c323ce2ecab04945b506bf04bddd43c7582c9fbf98ce8b","f380f754917bdc6d17093f871ff43201664a75572d28b860df518fdbf8f740be","61ad1ec61d505cf37d94dca60e8e997a7ab9c9b37ac0937c447c6f99f5eaae1d","9f3e390e24563480e463d1bb5df8d5896e544ecb25dba17b3843741b6d6890d1","035dc87f090626f361ac7a1fe771c647be6d32cf29bb0b110a6bf346deae287f","c3c59ba667c814a640c01ea119d156347dc04c3844e0067f4e60f9ffae877e8f","197bbcf0b10e2a83a3b3bac20fa5140966e6e15fa8a10b0b238c50bc66a87d66","4c30199ad6fdba1851b0f0e0521ee8b521206bf1d70413dad7404cb0cedbb3e3","530f5f83c26d9065c72fab1d85bb3ea6ed8301d7b9faa7374e7e2296bc6ae626","0c0a3178d6df74468b152130b4f03cb06ce82923df444d678aa13d5de0666814","36a88d8b6dff9f0081a9fe40a5638ea5c9411372182ccf338b1a59568efe35d3",{"state":1171,"reviewer":1172,"selfReview":1173,"reviewedAt":1174,"method":1175},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899599056]