---
title: "Symmetry"
book: "Primary & Middle School Mathematics"
subject: math
language: en
chapter: 27
exercises: 11
source: https://one-course.com/books/math/1/en/chapter/27-symmetry
---

# Chapter 27 — Symmetry

Butterflies, [faces](https://one-course.com/books/math/1/en/chapter/11-shapes-and-solids#def-g2-shapes-solids), snowflakes: nature loves figures whose two halves match when folded. This chapter explores symmetry with folds and grid paper — constructions with the set square wait for [Chapter 42](https://one-course.com/books/math/1/en/chapter/42-axial-symmetry#ch-g6-symmetry).

## 27.1 Folding and matching

**Definition 27.1 (Axis of symmetry).**

A line is an *axis of symmetry* of a figure when folding the figure along that line makes the two halves match *exactly*. A figure can have no axis, one axis, or several.

![Fold along the dashed line: the halves match. The last triangle has no fold that works — no axis of symmetry.](https://one-course.com/images/onecourse/chapters/math-1/g4-symmetry/fig-84456442da6f.svg)

*Fold along the dashed line: the halves match. The last triangle has no fold that works — no [axis of symmetry](#def-g4-symmetry-def).*

**Example 27.2 (Testing with tracing paper).**

Trace the figure, flip the tracing paper over along the candidate axis, and lay it back: if the tracing covers the figure exactly, the axis is genuine. Eyes alone are often fooled — a rectangle’s diagonal *looks* like an axis, but the fold test says no (try it!).

## 27.2 Completing figures on a grid

**Method 27.3 (Symmetric of a figure across a grid line).**

The axis is a vertical (or horizontal) line of the grid.

1. Take each [vertex](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) of the figure in turn;
2. count its distance to the axis in squares;
3. place the image [vertex](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) at the *same* distance on the *other* side, on the same row (or column);
4. join the image [vertices](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) in the same order, and check: the image is the mirror twin of the figure.

![Completing across the red axis: each vertex jumps to the same distance on the other side. Vertices touching the axis would stay put.](https://one-course.com/images/onecourse/chapters/math-1/g4-symmetry/fig-7196aa6feb76.svg)

*Completing across the red axis: each [vertex](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) jumps to the same distance on the other side. [Vertices](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon) touching the axis would stay put.*

**Example 27.4 (Same shape, same size, flipped).**

The image has exactly the same lengths and the same angles as the original: symmetry copies the figure — but flips it over, like a left hand and a right hand. If the original letter is F, its mirror image is a backwards F, not another F.

## 27.3 Axes of the usual shapes

**Example 27.5 (Counting axes).**

By folding:

- a square: $4$ axes (two through the middles of opposite sides, two along the diagonals);
- a rectangle: $2$ axes (middles of opposite sides only!);
- an isosceles triangle: $1$ ; an equilateral triangle: $3$ ;
- a circle: every line through its center — more axes than one can count.

**Example 27.6 (Symmetry in the alphabet).**

A, M, T, U, V have a vertical axis; B, C, D, E have a horizontal one; H, I, O, X have both. Words made of the right letters can be symmetric too: MUM has a vertical axis; OXO works both ways.

## 27.4 Exercises

**Exercise 27.1 ★.**

Trace and fold: which of these have an [axis of symmetry](#def-g4-symmetry-def) — a square; a (non-square) rectangle; a scalene triangle (all sides different); a circle?

**Solution of Exercise 27.1.**

Square: yes ($4$ axes). Rectangle: yes ($2$). Scalene triangle: no fold works. Circle: yes — any line through the center.

**Exercise 27.2 ★.**

Draw the axes of symmetry of: a square; a rectangle; an equilateral triangle. How many for each?

**Solution of Exercise 27.2.**

Square: $4$ (two middle lines, two diagonals). Rectangle: $2$ (the middle lines only). Equilateral triangle: $3$ (one through each [vertex](https://one-course.com/books/math/1/en/chapter/26-lines-and-polygons#def-g4-geometry-polygon)).

**Exercise 27.3 ★.**

Use the tracing-paper test of [Example 27.2](#ex-g4-symmetry-trace) to show that the diagonal of a $6 \times 4$ rectangle is *not* an [axis of symmetry](#def-g4-symmetry-def).

**Solution of Exercise 27.3.**

Folding along the diagonal, the two triangular halves have the same shape but do not lie on each other — one is “turned the wrong way”: the tracing lands beside the figure, not on it. So the diagonal is not an axis.

**Exercise 27.4 ★.**

On grid paper, draw a vertical axis and the letter L (three squares tall, two wide) at $2$ squares from the axis. Construct its mirror image.

**Solution of Exercise 27.4.**

The image L is $2$ squares on the *other* side of the axis, written backwards (a mirror L), same sizes.

**Exercise 27.5 ★.**

On grid paper, draw a horizontal axis and a small flag (a $1 \times
1$ triangle on a $3$-square mast) above it. Complete the figure by symmetry below the axis.

**Solution of Exercise 27.5.**

The completed figure shows the flag and its upside-down reflection below the axis, mast under mast.

**Exercise 27.6 ★.**

A figure touches its [axis of symmetry](#def-g4-symmetry-def) at a point $P$. Where is the image of $P$? Explain with the folding picture.

**Solution of Exercise 27.6.**

$P$ does not move: folding along the axis leaves every point *on* the axis exactly where it is — $P$ is its own image.

**Exercise 27.7 ★.**

Sort the capital letters A, B, E, F, H, N, O, T: vertical axis? horizontal axis? both? neither?

**Solution of Exercise 27.7.**

Vertical axis: A, H, O, T. Horizontal axis: B, E, H, O. Both: H, O. Neither: F, N.

**Exercise 27.8 ★.**

A triangle has sides $5$ cm, $5$ cm and $3$ cm. Does it have an [axis of symmetry](#def-g4-symmetry-def)? Which sides does the fold exchange?

**Solution of Exercise 27.8.**

Yes: the triangle is isosceles, and its axis passes through the tip between the two $5$ cm sides and the middle of the $3$ cm base. The fold exchanges the two equal sides.

**Exercise 27.9 ★★.**

On grid paper, draw a figure of your own that has *exactly two* axes of symmetry, and draw both axes. (Hint: think of a rectangle, or of a cross.)

**Solution of Exercise 27.9.**

Any figure with exactly two axes works: a (non-square) rectangle, a plus-sign with arms of two different lengths (long horizontal, short vertical), an oval. The two axes always cross at the center of the figure.

**Exercise 27.10 ★★.**

[Half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of a symmetric drawing is given: the left [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) of a house (a square wall and [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) a triangular roof), with a vertical axis along its right edge. Describe or draw the complete house. How do the lengths of the completed right [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) compare to the left [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half)?

**Solution of Exercise 27.10.**

The complete house is the left [half](https://one-course.com/books/math/1/en/chapter/17-sharing-and-division#def-g3-division-half) plus its mirror twin: a wall twice as wide as the half-wall, under a full triangular roof. Every length on the right equals its partner on the left — symmetry copies lengths exactly.

**Exercise 27.11 ★★.**

Aya claims: “every triangle with two equal sides has an [axis of symmetry](#def-g4-symmetry-def), and every triangle with an [axis of symmetry](#def-g4-symmetry-def) has two equal sides.” Test her double claim on drawings: an isosceles triangle, an equilateral one, a scalene one. Does the folding support her?

**Solution of Exercise 27.11.**

The drawings support both directions: the isosceles triangle folds along the line through its tip (two equal sides $\to$ axis); the equilateral one folds three ways; the scalene one has no axis (no equal sides $\to$ no axis). Aya’s double claim is right — it will be proved in [Chapter 42](https://one-course.com/books/math/1/en/chapter/42-axial-symmetry#ch-g6-symmetry).