PrepShorts · Study sheet · Class 8 Mathematics · Chapter 4, Quadrilaterals
Chapter 4 · Quadrilaterals
The carpenter's problem: how to be sure a frame really is rectangular
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A carpenter making a rectangular frame never has to measure a corner. That is not a saving of effort — it is the only way to be sure.
The idea
A carpenter who wants a rectangular frame never has to measure a single corner. Cut two strips of the same length, pin them together at the point that halves both, run a thread round the four ends, and the four corners come out square whatever angle the strips happen to cross at. The reason is a cancellation: the crossing angle x fixes the angles at the pin, each of the four triangles round the pin is isosceles because all four half-strips are the same length, and each corner of the frame is one base angle from each of two neighbouring triangles — (90 − x/2) added to x/2. The x disappears. That vanishing x is the whole video: it is why the joint can be sloppy and the frame still true.
What you should be able to do
- State the carpenter's three questions about the two strips — length, joining point, crossing angle — and say which of the three turns out not to matter
- Prove that a rectangle's two diagonals are the same length, by finding the two triangles that share a side and applying SAS
- Prove that a rectangle's diagonals cut each other in half, by showing the two angles either side of the crossing are equal and applying AAS
- Use the word bisect correctly for a segment and for an angle
- Work every angle in a figure whose equal diagonals bisect each other at 60°, starting from the isosceles triangles at the crossing point
- Repeat that work with a general crossing angle x and show that each corner comes to 90° with x cancelling out
- Explain why this converse is the statement the carpenter actually needs, and why the forward statement alone would not build the frame
- Describe the joint that produces a rectangle from one 8 cm strip, giving the second strip's length and the position of the pin
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| quadrilateral | a closed figure made of four straight sides | printed on the chapter's opening page (Part I, p.82) |
| rectangle | a quadrilateral whose corners are all right angles | defined in this chapter (Part I, §4.1, p.83) |
| diagonal | a segment joining two corners that are not next to each other | printed in this chapter (Part I, §4.1, p.83) |
| bisect | to cut into two equal parts — used here of a segment and of an angle | printed in this chapter (Part I, §4.1, p.85) |
| midpoint | the point of a segment that has equal lengths on either side | printed in this chapter (Part I, §4.1, p.85) |
| congruence | the relation between two figures that match part for part | printed in this chapter (Part I, §4.1, p.84); carried in from Class 7 |
| SAS | the congruence condition using two sides and the angle between them | printed in this chapter (Part I, §4.1, p.84) |
| AAS | the congruence condition using two angles and a side not between them | printed in this chapter (Part I, §4.1, p.85) |
| vertically opposite angles | the equal pair formed on opposite sides of a crossing | printed in this chapter (Part I, §4.1, p.84) |
| linear pair | two angles side by side on a straight line, totalling 180° | printed in this chapter (Part I, §4.1, p.85) |
| isosceles | having two sides of equal length | printed in this chapter (Part I, §4.1, p.87) |
| base angles | the two equal angles facing the equal sides of an isosceles triangle | printed in this chapter (Part I, §4.1, p.87) |
| deduction | working a new fact out from facts already settled, rather than measuring | printed in this chapter as the heading of each numbered argument (Part I, §4.1, p.84) |
| converse | the statement got by swapping what is assumed with what is concluded | an added term; not printed in this chapter, which performs the swap without naming it |
Where people slip up
- "You need a protractor to get a right angle." The entire point is that you do not. Two equal sticks and a thread will do it, which is why the chapter asks for exactly that at Part I p.94, exercise 4.
- "Equal diagonals are enough." They are not. A shape whose diagonals match in length but do not cut each other in half need not have a single right angle. Both conditions are doing work.
- "Diagonals that bisect each other are enough." Also not. That gives the corners equal in opposite pairs but not necessarily square — the parallelogram of §4.3 is exactly this case, and it is worth foreshadowing here.
- "The crossing angle must be 90°." Students transfer this from the square. At 90° you get a square only if the diagonals are also equal; at any other angle, with equal bisecting diagonals, you still get a rectangle. The angle is the one input that is free.
- "We proved rectangles have these diagonals, so we're done." The carpenter is standing at the other end of the argument: she has the diagonals and wants the rectangle. Deductions 1 and 2 go one way, Deduction 3 goes the other.
- "I built five and they all came out right, so it works." The chapter warns against exactly this at Part I p.88 with its thousandth-rectangle question. It belongs in this topic as a caution and is taken up properly in the next one.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 4.1 Q1, Figure it Out · 4.1 Q4
Transcript1,276 words
You are making a rectangular frame, and the only thing that matters is that the corners come out square. The obvious way to check is to measure a corner. Put a set square in it and look. But a corner is exactly where a frame is hardest to measure. There is a joint there, there is glue, and there is usually a nail. And measuring one corner tells you about one corner.
So here is a way to build the frame that never measures a corner at all, and cannot go wrong. It is two sticks and a pin, and the reason it works is a piece of arithmetic where something disappears. That disappearance is the whole of this. Cut two strips of wood the same length. Eight centimetres each, say. Pin them together at the point that halves both of them. Four centimetres either side of the pin, on both strips.
That is the entire specification. Two rules: same length, pinned at the middle. Now open them out to any angle you like. Any angle at all. Run a thread around the four loose ends, and the four ends are the corners of your frame. The claim is that those four corners are square. Not roughly square. Square. And nobody measured anything. Take the strips crossing at sixty degrees and check it.
The four angles where they cross are sixty, one hundred and twenty, sixty, one hundred and twenty. Join the four ends. Measure the corner at each end of the thread. Ninety. Ninety. Ninety. Ninety. Which is a surprise, because sixty degrees is nowhere in that answer. You could reasonably think sixty was lucky. A neat angle, a neat result. So the thing to do is try it at an angle nobody would ever choose.
Open the strips to twenty-three degrees. All four corners: ninety. Open them to one degree, so the frame is a splinter. All four corners: ninety. So try every whole angle the strips can cross at, from one degree to a hundred and seventy-nine. That is a hundred and seventy-nine frames, and in every single one of them all four corners are square. Not one fails. Try it again in half-degree steps and three hundred and fifty-eight frames come out square.
Across all of those, the corner takes exactly one value. So the crossing angle is not just unimportant. It is completely absent from the answer, and that has to be explainable. Call the crossing angle x, and follow it. At the pin, x makes four angles: the narrow one is x, and next to it is what is left of a straight line. So the four angles at the pin are x, one hundred and eighty minus x, x again, and one hundred and eighty minus x again.
x is very much present there. It is doing everything. Now look at what those four angles are the apexes of. The pin splits the frame into four triangles, and each of them has one of those angles at its tip. The question is what happens to x on the way from the tip of a triangle to its base. This is where the two rules start earning their keep.
Same length, pinned at the middle. So all four half-strips are the same length. Every one of those four triangles has two of those half-strips as its two sides. Which makes every one of them isosceles — two equal sides, and so two equal angles facing them. That is not a coincidence of the drawing. It is the two rules, restated. And an isosceles triangle is a machine for turning an apex angle into two equal base angles.
Feed it x and see what comes out. Take a narrow triangle first. Its tip is x, and its two base angles are equal — call each one a. The three angles of any triangle come to one hundred and eighty. So a plus a plus x is one hundred and eighty. Which gives a equal to ninety minus half of x. Now a wide triangle. Its tip is one hundred and eighty minus x, and its equal base angles are each b.
b plus b plus one hundred and eighty minus x is one hundred and eighty, so b is half of x. At sixty degrees that is a equals sixty and b equals thirty, which is exactly what the drawing shows. Both of those still have x in them. Both of them change when you open the strips. And now the step the whole thing rests on. Look at one corner of the frame — one of the four ends of the thread.
Two triangles meet there. The corner is one base angle from each of them. So the corner is a plus b. That is ninety minus half of x, plus half of x. The halves of x cancel. Ninety. And there is no x left in it to argue with. The angle enters twice, once being subtracted and once being added, and it leaves. Two rules went in. It is worth knowing what each of them was for.
Break the first one. Pin at the middle of both, but make the second strip shorter. Run the sweep again. Of the hundred and seventy-nine angles, exactly none give a square corner. But look at what you do get. Opposite sides still match, and opposite corners are still equal to each other. It is a perfectly good parallelogram. It is simply not a rectangle. Which is worth pausing on, because matching opposite sides is what most people would check.
Matching sides is not the test. It survives the mistake. Now break the other rule instead. Two strips the same length, but pin them off-centre. Sweep again: of the hundred and seventy-nine, none are square either. And this time the damage is worse. The opposite sides stop matching as well. The four corners come out all different from each other. Nothing is left. So the two rules are not one rule said twice. They fail in different directions.
Equal lengths is what makes the frame a rectangle rather than a leaning parallelogram. Pinning at the middle is what makes it a parallelogram in the first place. One more thing, and it is the reason the whole method is worth having. You might still want to just check a corner at the end and be done. Slide the pin a hundredth of a centimetre off centre. One hundredth. Every corner of that frame is within a fifth of a degree of square.
You cannot see a fifth of a degree. No set square you own will catch it. And the frame is wrong. Worse: make three half-strips equal and one longer, and one corner comes out exactly ninety while the other three do not. So a frame can pass the corner you happen to check and fail the three you did not. Which is why the strips are better than the set square.
The two rules are both about LENGTH, and length is the one thing a workshop can get right. You can cut two sticks the same. You can find the middle of a stick. Neither needs a protractor. And then the angle at the pin can be anything, because the arithmetic throws it away. The joint can be sloppy. It can shift while you are working. The frame is still true.
That is not a trick. It is what it looks like when a quantity enters an argument twice with opposite signs. Somebody worked that out once, and carpenters have been squaring frames with two sticks and a pin ever since.
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.
Comes up again in
- How a property gets deduced, and why you still check it against a real shapeClass 8 · Ch 4, Quadrilaterals
- Properties of a rectangle, and why the square is the special caseClass 8 · Ch 4, Quadrilaterals
- What the diagonals alone tell you about a quadrilateralClass 8 · Ch 4, Quadrilaterals
- What happens to a product when you nudge one factorClass 8 · Ch 6, We Distribute, Yet Things Multiply
Either side of this one
- The Hindu number system, and why treating 0 as a digit changed everythingClass 8 · Ch 3, A Story of Numbers