PrepShorts · Study sheet · Class 7 Mathematics · Chapter 7, A Tale of Three Intersecting Lines
Chapter 7 · A Tale of Three Intersecting Lines
Why a compass beats trial and error for building a triangle
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A compass is not a tidier ruler. That is the whole video.
The idea
A compass is not a tidier ruler. Marking a point with a ruler produces a point that happens to be 4 cm from A, and you then have to go and check whether it is also 4 cm from B — and it usually is not. An arc is different in kind: it is every point at that distance, all at once. So where two arcs cross, both distance conditions are already true, and no measuring afterwards can overturn it. That is why the construction terminates, and why the answer to "can you see why it works?" is the whole point rather than a footnote.
What you should be able to do
- Name the three vertices, three sides and three angles of a labelled triangle, and write the triangle's name in any vertex order
- Explain why a ruler-only construction of a triangle needs repeated trials while a compass construction does not
- State what the arc of radius 4 cm centred at A contains, and why that makes it the right tool
- Justify, without measuring the finished figure, why the crossing point of two arcs is at the correct distance from both base vertices
- Construct an equilateral triangle of a given sidelength using a compass
- Construct a triangle from three given sidelengths, choosing a base and working through the four printed steps
- Say why the full circles may be replaced by short arcs without weakening the argument
- Distinguish equilateral from isosceles by the number of equal sidelengths
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| triangle | the simplest closed figure: three corner points joined in pairs | printed throughout, from p.146 |
| vertex | one of the three corner points of a triangle | printed on p.146 |
| side | one of the three segments joining a pair of vertices | printed on p.146 |
| sidelength | the length of a side, written by this book as one word | printed on p.148 and used throughout §7.2 |
| marked ruler | a straight edge carrying a scale, so it can measure as well as rule | printed on p.147 |
| compass | the two-armed instrument that carries a fixed distance round a point | printed on p.147 |
| arc | part of a circle: the points at one fixed distance from a centre | printed on p.147 |
| radius | the fixed distance from the centre to any point of the circle or arc | printed on p.147 |
| circle | every point at one fixed distance from the centre | printed on p.149 |
| centre | the point a circle or arc is drawn about | printed on p.149 |
| base | the side drawn first, chosen by you rather than given by the problem | printed on p.147 |
| point of intersection | the point where two drawn curves cross | printed on p.148 and again on p.150 |
| equilateral triangle | a triangle whose three sides are of equal length | printed as the §7.1 heading, p.146, and again in the text on p.150 |
| isosceles triangle | a triangle with two sides of equal length | printed in bold on p.150 |
Two cautions: The book writes sidelength as one word and uses it constantly; keep it, because "length of the side" invites the student to think the length is a separate object from the side. And this topic never needs the word congruent — that word is not printed anywhere in this chapter (all twenty-seven printed pages were checked).
Where people slip up
- "The compass is just a neater way of drawing." It is a different claim. A ruler mark asserts one distance; an arc asserts a whole set of points, so the crossing satisfies two conditions simultaneously. Section 7 must make the student able to say why no check is needed afterwards.
- "A triangle has to sit on its longest side / on a horizontal side." The base is a choice made by the person constructing, not a property of the triangle. In the 4-5-6 construction the chapter puts the 4 cm side at the bottom purely for convenience.
- "∆ABC and ∆BAC are different triangles." They are the same triangle named two ways; p.146 says the vertex order is free, and letters its examples out of order to prove it.
- "Two arcs always cross, so any three lengths work." They do not, and the whole of The triangle inequality: which three lengths can close is about when they fail. Do not let the explanation imply that the method is universal — it should end pointing at the question, exactly as §7.2 does.
- "Isosceles means exactly two equal sides, so an equilateral triangle is not isosceles." The chapter's phrasing is "two equal sides", not "exactly two", and it derives isosceles triangles from a circle's radii — where all radii are equal. Treat "exactly two" as a convention some books use and this one does not state either way.
- "The dashed circle in the figure is part of the answer." The dashes record what the compass could reach. Only the crossing point is used.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 7.2 Q1, Figure it Out · 7.2 Q2
Transcript1,333 words
Three points. Join each pair with a straight segment. That is a triangle — the simplest closed figure there is. Three corners, which we call vertices. Three segments, which we call sides. And three angles, one at each corner, which you get for free the moment you join the points. Here are four of them, and they look nothing like each other. Tall and thin. Wide and flat. Leaning one way, leaning the other.
All four are triangles, because all four are three points joined in pairs. Nothing says a triangle has to sit flat, or look balanced, or point upwards. Now, the name. Label the corners A, B and C, and the triangle is called A B C. But you could just as well have started at B. Or at C. B C A. C A B. A C B. Six different orders altogether, and every one of them names the same three corners.
So triangle A B C and triangle B A C are not two triangles. They are one triangle, written down two ways. Worth knowing before you meet a question that names one the other way round and you decide it is a different shape. The letters are labels. Move them about all you like — the shape does not care. One thing before we start building. What if the three points sit on a straight line?
Put one here. One two centimetres along. One five centimetres from the first. Join them in pairs. You get three segments — two, three and five. But there is no inside. No area. Nothing enclosed. It is a line with two extra dots on it. And notice why. Two plus three is exactly five. The two shorter pieces, laid end to end, reach precisely as far as the long one, with nothing left over to lift a corner off the line.
Hold that thought. It matters more than it looks. Right. Let us actually build one. Start with the most symmetric case there is. All three sides four centimetres. The base first. Draw a segment, measure it, four centimetres. Call the ends A and B. That part is easy. A ruler does it perfectly. Now we need a third corner. Call it C. And C has two jobs. It has to be four centimetres from A, and four centimetres from B.
Two conditions, and one point that has to satisfy both of them. Here is where most people reach for the ruler again, and here is where it goes wrong. Watch the ruler attempt. Line the ruler up from A, in some direction, and mark a point four centimetres along. That point is four centimetres from A. Job one, done. Now measure from B to it. Six point two. Not four.
So rub it out, tilt the ruler, mark another point four centimetres from A. Measure again. Three point one. Again. Five point four. Again. Two point eight. You can do this all afternoon. Every mark satisfies the first condition and misses the second, and nothing in the method tells you which way to tilt next. So stop, and change what you are drawing. Open a compass to four centimetres. Put the point on A.
Now sweep. Look at what that curve is. Every point on it is four centimetres from A. Not one point. All of them, at once, and you did not measure a single one. That is the difference, and it is not a difference of neatness. A ruler mark says: this particular point is four centimetres from A. An arc says: here is every point that is. One of those is an answer. The other is the whole set of candidates, drawn in a single sweep.
Now do it again from B. Compass still open to four. Point on B. Sweep. Every point on this second curve is four centimetres from B. And the two curves cross. Look at that crossing. It is on the first curve, so it is four centimetres from A. It is on the second curve, so it is four centimetres from B. Both conditions true at the same time — and neither was checked afterwards. They were true the moment the arcs were drawn.
Join it to A and to B. Three sides, all four centimetres. Done, with nothing left to verify. Now the general case, where the sides are not all the same. Four centimetres, five centimetres, six centimetres. First decision. Which side goes at the bottom? And the answer is, whichever you like. The base is your choice, not the puzzle's. It is not a property of the triangle. It is the side you happened to draw first.
Take the four. Draw it, label the ends A and B. Now C has to be five centimetres from A, and six centimetres from B. The same two conditions as before, with different numbers in them. Compass to five. Point on A. This time draw the whole circle, so you can see what you are claiming. Every point on that circle is exactly five centimetres from A. Every one, all the way round.
Compass to six. Point on B. Whole circle again. Every point on this one is six centimetres from B. And the two circles cross. Twice, in fact — once above the line and once below. Take the upper one. That is C. You do not need the whole of either circle. Only the little piece near the crossing does any work. The rest is a record of where the compass could have gone. Rub it away, or never draw it, and the argument is untouched.
Let us be precise about why C is right, because this is the part people skip. C is on the circle centred at A with radius five. That is what being on that circle means. So A C is five centimetres. C is on the circle centred at B with radius six. So B C is six centimetres. And A B is four, because we drew it that way with a ruler.
Four, five and six. All three, and I have not measured the finished figure once. Compare that with the ruler attempt, where you had to measure at the end to find out whether you had failed. Here there is nothing to find out. The distances are true because of the way the point was made. So here is the whole method, in four steps. One. Pick a side to be the base. Draw it with the ruler, and label the ends.
Two. Open the compass to the second sidelength, point on one end, draw an arc. Three. Open the compass to the third sidelength, point on the other end, and draw an arc that crosses the first. Four. That crossing is your third corner. Join it to both ends. That is it. No trial. No measuring at the end. And it works for the reason we spent so long on. Each arc carries a whole distance condition, so the crossing carries both.
Four steps that finish, instead of a tilt-and-measure that never does. One more idea, and it comes free with what we have built. Draw a circle. Mark the centre. Now pick any two points on it, and join both to the centre. Those two segments are radii, so they are equal. Every time. Whichever two points you pick. A triangle with two equal sides is called isosceles. So a circle is a machine for making them. Pick any two points and you have another one.
If all three sidelengths are equal it is called equilateral, like the four-four-four we started with. And one last thing to leave you with. Try three lengths of two, three and five. Draw the base of five, sweep an arc of two, sweep an arc of three, and watch what the arcs do. They do not cross. They touch. Which tells you the method has a limit — and finding it is where this goes next.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- The four angles at a crossing: vertically opposite and linear pairsClass 7 · Ch 5, Parallel and Intersecting Lines
Comes up again in
- The triangle inequality: which three lengths can closeClass 7 · Ch 7, A Tale of Three Intersecting Lines
- Constructing from two sides and the angle between themClass 7 · Ch 7, A Tale of Three Intersecting Lines
- What an altitude is, and how to construct oneClass 7 · Ch 7, A Tale of Three Intersecting Lines
- Two independent ways to classify a triangle: by side and by angleClass 7 · Ch 7, A Tale of Three Intersecting Lines
Either side of this one
- Cryptarithms: recovering digits from constraints aloneClass 7 · Ch 6, Number Play