---
title: "Triangles and Quadrilaterals"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 41
exercises: 12
source: https://one-course.com/books/math/1/en/chapter/41-triangles-and-quadrilaterals
---

# Chapter 41 — Triangles and Quadrilaterals

Triangles and [quadrilaterals](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) are the basic shapes of geometry. The goal of this chapter is to know their special families — [isosceles](#def-g6-shapes-triangles), [equilateral](#def-g6-shapes-triangles), [right triangles](#def-g6-shapes-triangles); rectangles, rhombuses, squares — by their *defining properties*, and to construct them precisely with ruler and compass.

## 41.1 Triangles

**Definition 41.1 (Special triangles).**

A triangle $ABC$ has three [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) and three sides. It is:

- *isosceles* at $A$ when $AB = AC$ (two equal sides);
- *equilateral* when $AB = BC = CA$ (three equal sides);
- *right-angled* at $A$ when the angle $\widehat{BAC}$ is a [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) . The side $[BC]$ opposite the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) is called the *hypotenuse* .

![The three special triangles. Small tick marks indicate equal sides; the square marks the right angle.](https://one-course.com/images/onecourse/chapters/math-1/g6-shapes/fig-ba6b18b61d3e.svg)

*The three special triangles. Small tick marks indicate equal sides; the square marks the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle).*

**Method 41.2 (Constructing a triangle from its three sides).**

To construct $ABC$ with $AB = 5$ cm, $AC = 4$ cm, $BC = 3$ cm:

1. draw the [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[AB]$ of $5$ cm with the ruler;
2. draw an arc of the circle of center $A$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm (all points at $4$ cm from $A$ );
3. draw an arc of the circle of center $B$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $3$ cm;
4. the point $C$ is where the two arcs cross; join it to $A$ and $B$ .

![Constructing a triangle with the compass: the vertex C is the crossing point of two arcs. (In you will see when such a construction is possible.)](https://one-course.com/images/onecourse/chapters/math-1/g6-shapes/fig-df026e474b4b.svg)

*Constructing a triangle with the compass: the [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) $C$ is the crossing point of two arcs. (In [Chapter 51](https://one-course.com/books/math/1/en/chapter/51-triangles-and-angles#ch-g7-triangles) you will see when such a construction is possible.)*

## 41.2 Quadrilaterals

**Definition 41.3 (Special quadrilaterals).**

A [quadrilateral](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) has four [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) and four sides. It is a:

- *rectangle* when its four angles are [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) ;
- *rhombus* when its four sides have the same length;
- *square* when it is both: four [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) *and* four equal sides.

**Proposition 41.4 (Properties of the diagonals).**

In these [quadrilaterals](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon), the two diagonals cut each other at their common midpoint, and moreover:

- in a rectangle, the diagonals have the *same length* ;
- in a [rhombus](#def-g6-shapes-quadrilaterals) , the diagonals are *[perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp)* ;
- in a square, both at once.

**Proof.** *Admitted at this level.* ∎

![The three special quadrilaterals with their diagonals (in red). The square inherits the properties of both the rectangle and the rhombus.](https://one-course.com/images/onecourse/chapters/math-1/g6-shapes/fig-565eb595b668.svg)

*The three special [quadrilaterals](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) with their diagonals (in red). The square inherits the properties of both the rectangle and the [rhombus](#def-g6-shapes-quadrilaterals).*

**Remark 41.5.**

A square is automatically a rectangle (it has four [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle)) and automatically a [rhombus](#def-g6-shapes-quadrilaterals) (four equal sides). So “rectangle” does not mean “not a square”! In mathematics, the more special shape belongs to the more general family.

**Example 41.6 (Recognizing a shape from its properties).**

A [quadrilateral](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) has four sides of length $4$ cm and one [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle). What is it? Four equal sides make it a [rhombus](#def-g6-shapes-quadrilaterals). In a [rhombus](#def-g6-shapes-quadrilaterals), opposite angles are equal and the four angles add up to $360^\circ$; one [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) then forces all four to be right (this will be proved with parallelograms in [Chapter 52](https://one-course.com/books/math/1/en/chapter/52-central-symmetry-and-parallelograms#ch-g7-central)). So the shape is a square.

**Method 41.7 (Construction programs).**

A *construction program* is a list of instructions that anyone can follow to reproduce a figure exactly. Write one instruction per line, name every point when it is created, and only use: “draw the [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) / line / circle…”, “place the point where … cross”. Test your program by following it literally, word for word.

**Example 41.8.**

Program for a square $ABCD$ of side $4$ cm:

1. Draw a [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[AB]$ with $AB = 4$ cm.
2. Draw the [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to $(AB)$ at $A$ ; on it, place $D$ at $4$ cm from $A$ (choose one side of the line and stick to it).
3. Draw the [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to $(AB)$ at $B$ ; on it, place $C$ at $4$ cm from $B$ , on the same side as $D$ .
4. Join $C$ and $D$ .

Check with the ruler: $DC$ should measure $4$ cm too.

## 41.3 Exercises

**Exercise 41.1 ★.**

Construct a triangle $ABC$ with $AB = 6$ cm, $AC = 5$ cm and $BC = 4$ cm, following [Method 41.2](#met-g6-shapes-construct).

**Solution of Exercise 41.1.**

Follow the method: [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[AB]$ of $6$ cm, arc of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $5$ cm centered at $A$, arc of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm centered at $B$; the crossing point is $C$.

**Exercise 41.2 ★.**

Construct an [isosceles triangle](#def-g6-shapes-triangles) $DEF$ with $DE = DF = 5$ cm and $EF = 3$ cm. Which two angles of this triangle look equal? Measure them to check.

**Solution of Exercise 41.2.**

Base $[EF]$ of $3$ cm, then two arcs of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $5$ cm centered at $E$ and $F$; their crossing is $D$. The two base angles $\widehat{DEF}$ and $\widehat{DFE}$ are equal (about $72.5^\circ$ each) — symmetry of the [isosceles triangle](#def-g6-shapes-triangles) ([Example 42.8](https://one-course.com/books/math/1/en/chapter/42-axial-symmetry#ex-g6-symmetry-proves)).

**Exercise 41.3 ★.**

Construct an [equilateral triangle](#def-g6-shapes-triangles) of side $4$ cm using only ruler and compass. Measure one of its angles.

**Solution of Exercise 41.3.**

[Segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) of $4$ cm, then two arcs of [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $4$ cm centered at its ends. Every angle measures $60^\circ$.

**Exercise 41.4 ★.**

Classify each triangle from its data ([isosceles](#def-g6-shapes-triangles), [equilateral](#def-g6-shapes-triangles), [right-angled](#def-g6-shapes-triangles) — several answers may apply): (a) sides $7$, $7$, $7$; (b) sides $5$, $5$, $8$; (c) angles $90^\circ$, $45^\circ$, $45^\circ$; (d) angles $60^\circ$, $60^\circ$, $60^\circ$.

**Solution of Exercise 41.4.**

(a) [equilateral](#def-g6-shapes-triangles) (hence also [isosceles](#def-g6-shapes-triangles)). (b) [isosceles](#def-g6-shapes-triangles) ($5 = 5$). (c) [right-angled](#def-g6-shapes-triangles) and [isosceles](#def-g6-shapes-triangles) (two equal angles of $45^\circ$ [face](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) two equal sides). (d) [equilateral](#def-g6-shapes-triangles) (three equal angles).

**Exercise 41.5 ★.**

In a [right triangle](#def-g6-shapes-triangles), which side is the hypotenuse? Draw a triangle $MNP$ [right-angled](#def-g6-shapes-triangles) at $N$, and name its hypotenuse. Which [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) does the hypotenuse never touch?

**Solution of Exercise 41.5.**

The hypotenuse is the side opposite the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle), always the longest side. In $MNP$ [right-angled](#def-g6-shapes-triangles) at $N$, the hypotenuse is $[MP]$: it never touches the [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle), $N$.

**Exercise 41.6 ★.**

True or false? Justify each answer in one sentence.

1. Every square is a rectangle.
2. Every rectangle is a square.
3. Every square is a [rhombus](#def-g6-shapes-quadrilaterals) .
4. A [rhombus](#def-g6-shapes-quadrilaterals) with a [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) is a square.

**Solution of Exercise 41.6.**

*1. True*: a square has four [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle), which is the definition of a rectangle.

*2. False*: a $5 \times 3$ rectangle has unequal sides.

*3. True*: a square has four equal sides, the definition of a [rhombus](#def-g6-shapes-quadrilaterals).

*4. True*: see [Example 41.6](#ex-g6-shapes-recognize).

**Exercise 41.7 ★.**

Construct a rectangle $ABCD$ with $AB = 5$ cm and $BC = 3$ cm. Measure its two diagonals: what do you observe?

**Solution of Exercise 41.7.**

Construction with [perpendiculars](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp), as in [Example 41.8](#ex-g6-shapes-program). Both diagonals measure about $5.8$ cm: the diagonals of a rectangle have the same length ([Proposition 41.4](#prop-g6-shapes-diagonals)).

**Exercise 41.8 ★.**

Construct a [rhombus](#def-g6-shapes-quadrilaterals) with diagonals of $6$ cm and $4$ cm. (Use [Proposition 41.4](#prop-g6-shapes-diagonals): draw the two diagonals first — [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp), crossing at their midpoints — then join the four ends.)

**Solution of Exercise 41.8.**

Draw a [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) of $6$ cm; its midpoint $O$; the [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) at $O$; on it, two points at $2$ cm from $O$ on either side. Joining the four endpoints gives the [rhombus](#def-g6-shapes-quadrilaterals) (all four [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) are at distances $3$ cm and $2$ cm from $O$ along [perpendicular lines](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp), so the four sides are equal by symmetry).

**Exercise 41.9 ★★.**

Write a construction program for an [isosceles triangle](#def-g6-shapes-triangles) with base $4$ cm and equal sides of $6$ cm, in the style of [Example 41.8](#ex-g6-shapes-program). Exchange with a classmate: can they follow it without seeing your figure?

**Solution of Exercise 41.9.**

One correct program: 1. draw a [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[BC]$ with $BC = 4$ cm; 2. draw the arc of center $B$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $6$ cm above $[BC]$; 3. draw the arc of center $C$ and [radius](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-shapes) $6$ cm above $[BC]$; 4. name $A$ the crossing point of the two arcs; 5. draw the [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[AB]$ and $[AC]$.

**Exercise 41.10 ★★.**

Draw a circle of center $O$ and one of its [diameters](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-circle) $[AB]$. Place any point $M$ on the circle and draw the triangle $AMB$. Measure the angle $\widehat{AMB}$. Move $M$ and measure again. What do you conjecture? (This surprising fact is proved in [Chapter 60](https://one-course.com/books/math/1/en/chapter/60-the-right-triangle-and-the-cosine#ch-g8-cosine).)

**Solution of Exercise 41.10.**

Whatever the position of $M$ on the circle, the angle $\widehat{AMB}$ measures $90^\circ$: the triangle $AMB$ is [right-angled](#def-g6-shapes-triangles) at $M$ whenever $[AB]$ is a [diameter](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-circle). The proof waits for [Chapter 60](https://one-course.com/books/math/1/en/chapter/60-the-right-triangle-and-the-cosine#ch-g8-cosine).

**Exercise 41.11 ★★.**

A [quadrilateral](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) has diagonals of the same length that cut each other at their common midpoint. Sort out which shape it must be, using [Proposition 41.4](#prop-g6-shapes-diagonals): rectangle, [rhombus](#def-g6-shapes-quadrilaterals) or square? (Careful: only one property is given.)

**Solution of Exercise 41.11.**

Diagonals that cut each other at their midpoint and have equal lengths: that is the property list of the *rectangle* ([Proposition 41.4](#prop-g6-shapes-diagonals)). Nothing forces the diagonals to be [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp), so it need not be a [rhombus](#def-g6-shapes-quadrilaterals) or a square — a rectangle is the right answer (a square would satisfy it too, but is not forced).

**Exercise 41.12 ★★★.**

On a $4 \times 4$ dotted grid, how many different squares can you draw whose corners are grid dots — counting also the “tilted” squares? Draw at least one tilted square and explain why its sides are equal.

**Solution of Exercise 41.12.**

With $4 \times 4$ dots there are $3 \times 3$ grid cells. Grid-aligned squares: sides $1$, $2$ or $3$ give $9 + 4 + 1 = 14$ squares. Tilted squares also exist: the small ones whose sides are diagonals of $1 \times 1$ cells (like $(1,0)$, $(2,1)$, $(1,2)$, $(0,1)$) — four of them; and the larger ones whose sides are diagonals of $2 \times 1$ rectangles, like $(1,0)$, $(3,1)$, $(2,3)$, $(0,2)$ — two of them. All four sides of such a square are equal because they are diagonals of identical rectangles. Total: $14 + 4 + 2 = 20$ squares. (Any count with a correct tilted example and the $14$ aligned ones is a success at this level.)

## 41.4 Problem: What the diagonals know

**Problem 41.1.**

Weekend problem — rebuilding every special quadrilateral from its diagonals alone, and counting the diagonals of any polygon

[Proposition 41.4](#prop-g6-shapes-diagonals) reads a shape and tells you about its diagonals. This problem plays the game *backwards*: given only two crossing [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) — the future diagonals — what shape appears when you join their four ends? You will build a complete “diagonal dictionary”, understand *why* the [rhombus](#def-g6-shapes-quadrilaterals) construction of [Exercise 41.8](#exo-g6-shapes-8) works, meet a shape that has no name yet, and finish by counting the diagonals of [polygons](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) with many more than four sides.

**Part I — The diagonal dictionary.** For each question, draw the two [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) as described, join their four ends in order, and identify the [quadrilateral](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) you get.

1. Two [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) of $6$ cm each, [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) , crossing at their common midpoint. Measure the four sides and the four angles of the resulting shape: what is it?
2. Two [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) of $8$ cm and $5$ cm, [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) , crossing at their common midpoint. What shape appears ( [Exercise 41.8](#exo-g6-shapes-8) )?
3. Two [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) of $7$ cm each — equal but *not* [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) — crossing at their common midpoint. Measure the sides and angles: what do you conjecture the shape to be?
4. Two [segments](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) of $8$ cm and $5$ cm, not [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) , crossing at their common midpoint. The shape you get is none of the three of [Definition 41.3](#def-g6-shapes-quadrilaterals) — describe what you see (which sides look equal? [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) ?). This [quadrilateral](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) receives its name in [Chapter 52](https://one-course.com/books/math/1/en/chapter/52-central-symmetry-and-parallelograms#ch-g7-central) .
5. Summarize questions 1–4 in a dictionary with four entries: diagonals *equal / [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) / both / neither* (always crossing at their midpoints) $\rightarrow$ name (or description) of the shape.

**Part II — Why the constructions work.**

6. In question 2, the two diagonals cut the shape into four [right triangles](#def-g6-shapes-triangles) around the crossing point. What are the two leg lengths of each of these triangles? Explain why all four triangles are exact copies of one another, and deduce why the four sides of the shape are equal — the secret of [Exercise 41.8](#exo-g6-shapes-8) .
7. Deduce a compass-only test: after building any shape from its diagonals, how can you check with the compass alone (no ruler measurements) that the four sides are equal?
8. Write a construction program ( [Method 41.7](#met-g6-shapes-program) ) for a square whose *diagonals* measure $6$ cm — not its sides.
9. In the square of question 8, is the side longer or shorter than $3$ cm — [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the diagonal? Measure to find out. One [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of the answer can already be *proved* : going from one corner to the next via the center is a $3 + 3$ cm detour, and a straight path is always shorter than a detour — so the side is shorter than $6$ cm. (Proving that it also beats $3$ cm, and computing it exactly, is a job for [Chapter 58](https://one-course.com/books/math/1/en/chapter/58-the-pythagorean-theorem#ch-g8-pythagoras) .)
10. A new entry for the dictionary: draw a [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[AC]$ of $8$ cm; on it, mark the point $H$ at $3$ cm from $A$ ; draw through $H$ the [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[BD]$ of $5$ cm with $H$ as *its* midpoint. Join $ABCD$ . This shape is a *kite* (one diagonal bisects the other, at a [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) ). Using corner triangles as in question 6, find which sides of a [kite](#pb-g6-shapes-1) are equal to which.

**Part III — Counting diagonals.** A *diagonal* of a [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) joins two [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) that are not neighbours.

11. Draw a [quadrilateral](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) , a [pentagon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) (five [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ) and a [hexagon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) (six [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) ), and draw all their diagonals. Count them: how many in each case?
12. Counting without drawing, for the [hexagon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) : from one [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) , how many diagonals leave? (To how many of the six [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) can it *not* be joined by a diagonal?) Multiplying by the six [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) counts every diagonal exactly twice — why? Recover your count of question 11. (The same double-counting appeared in the handshakes of [Problem 37.1](https://one-course.com/books/math/1/en/chapter/37-whole-numbers#pb-g6-wholes-1) .)
13. Use the method of question 12 to count the diagonals of a [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) with $10$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) , then with $20$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) .
14. A [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) has exactly $44$ diagonals. How many [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) does it have? (Try the method of question 13 on a few candidates.)
15. Check your method on the smallest [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) of all: how many diagonals does a triangle have, and what does the counting method of question 12 answer? Conclude with one sentence on what handshakes and diagonals have in common.

**Solution of Problem 41.1.**

**1.** The four sides all measure the same (about $4.2$ cm) and the four angles are right: the shape is a *square*. (Equal diagonals give the [right angles](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle), as in the rectangle; [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) ones give the equal sides, as in the [rhombus](#def-g6-shapes-quadrilaterals); both at once give the square, [Proposition 41.4](#prop-g6-shapes-diagonals) read backwards.)

**2.** A *[rhombus](#def-g6-shapes-quadrilaterals)*: four equal sides (about $4.7$ cm), but no [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) — exactly the construction of [Exercise 41.8](#exo-g6-shapes-8).

**3.** The four angles come out right but the sides come in two different lengths: a *rectangle* — this is the situation of [Exercise 41.11](#exo-g6-shapes-11).

**4.** No equal-length claim survives measuring and no [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) appears; but opposite sides are equal two by two and look [parallel](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp). It is the general “slanted rectangle” whose diagonals merely bisect each other — the *parallelogram*, named and studied in [Chapter 52](https://one-course.com/books/math/1/en/chapter/52-central-symmetry-and-parallelograms#ch-g7-central).

**5.** Diagonals crossing at their common midpoint:

| diagonals… | shape |
| --- | --- |
| equal and [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) | square |
| [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) only | [rhombus](#def-g6-shapes-quadrilaterals) |
| equal only | rectangle |
| neither | the nameless one (see [Chapter 52](https://one-course.com/books/math/1/en/chapter/52-central-symmetry-and-parallelograms#ch-g7-central)) |

**6.** Each corner triangle has one leg along each diagonal: [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $8$ cm and [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of $5$ cm, so legs of $4$ cm and $2.5$ cm, with a [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) between them — the same data for all four. Triangles built from the same two legs and the [right angle](https://one-course.com/books/math/1/en/chapter/18-shapes-and-right-angles#def-g3-shapes-rightangle) between them are exact copies of one another (construct them: the compass settings are identical). Their four hypotenuses — the sides of the shape — are therefore equal: the shape is a [rhombus](#def-g6-shapes-quadrilaterals).

**7.** Open the compass to the length of one side, then walk it around the shape: if its point and pencil land exactly on each pair of neighbouring corners, all four sides equal that one opening. No ruler needed.

**8.** Program:

1. Draw a [segment](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-objects) $[AC]$ with $AC = 6$ cm and place its midpoint $O$ (measure $3$ cm from $A$ ).
2. Draw the [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp) to $(AC)$ at $O$ .
3. On it, place $B$ and $D$ at $3$ cm from $O$ , one on each side.
4. Join $A$ , $B$ , $C$ , $D$ .

Diagonals equal ($6$ cm), [perpendicular](https://one-course.com/books/math/1/en/chapter/40-lines-circles-and-angles#def-g6-lines-perp), bisecting each other: by the dictionary, $ABCD$ is a square.

**9.** Measuring gives about $4.2$ cm: *longer* than $3$ cm — the common guess “[half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) the diagonal” is wrong. The provable [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half): to go from corner $A$ to the next corner $B$, one can walk the side $[AB]$ straight, or take the detour $A \to O \to B$ through the center, costing $3 + 3 = 6$ cm; the straight side is shorter, so the side measures less than $6$ cm. That it is also *more* than $3$ cm is what the ruler shows; [Chapter 58](https://one-course.com/books/math/1/en/chapter/58-the-pythagorean-theorem#ch-g8-pythagoras) will prove it and give the exact value ($3$ cm multiplied by the square root of $2$, about $4.24$ cm).

**10.** The four corner triangles at $H$ now come in *two* matching pairs: two with legs $3$ cm and $2.5$ cm (hypotenuses $[AB]$ and $[AD]$), two with legs $5$ cm and $2.5$ cm (hypotenuses $[CB]$ and $[CD]$). So $AB = AD$ and $CB = CD$: a [kite](#pb-g6-shapes-1) has two pairs of equal *neighbouring* sides (the [rhombus](#def-g6-shapes-quadrilaterals) is the special [kite](#pb-g6-shapes-1) where all four match).

**11.** [Quadrilateral](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon): $2$ diagonals. [Pentagon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon): $5$. [Hexagon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon): $9$.

**12.** From one [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) of the [hexagon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon), diagonals go to every [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) except itself and its two neighbours: $6 - 3 = 3$ diagonals. Six [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) launch $6 \times 3 = 18$ diagonal-ends — but each diagonal has been counted at *both* of its ends, once from each endpoint, exactly like a handshake counted by both shakers ([Problem 37.1](https://one-course.com/books/math/1/en/chapter/37-whole-numbers#pb-g6-wholes-1)). True count: $18 \div 2 = 9$. It matches the drawing.

**13.** Ten [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def): $10 - 3 = 7$ from each, so $10 \times 7 \div 2 = 35$ diagonals. Twenty [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def): $20 \times 17 \div 2 = 170$ diagonals.

**14.** Try candidates: $10$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) give $35$ (question 13); $11$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) give $11 \times 8 \div 2 = 44$. Found: the [polygon](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) has $11$ [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def).

**15.** A triangle has no diagonals at all: every [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def)’s “non-neighbours” are … nobody, since the other two [vertices](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def) are both its neighbours. The method agrees: $3 - 3 = 0$ from each [vertex](https://one-course.com/books/math/1/en/chapter/35-cubes-and-boxes#def-g5-solids-def), total $0$. Handshakes and diagonals are the same count in different clothes: pairs chosen from a group, each pair counted once — mathematics recycles its good ideas.