Similar Figures

Similar figures have the same shape but not necessarily the same size. Check both conditions (equal angles and sides in the same ratio), find the scale factor, and name similar figures in matching order.

Before you start

Two ratios are equal when they simplify to the same ratio: $6 : 4 = 3 : 2$ and $9 : 6 = 3 : 2$ . As fractions, $\frac{9}{6} = \frac{6}{4} = 1.5$ .
Congruent figures have the same shape and size, and are named in matching order: in $\triangle ABC \equiv \triangle PQR$ , $A$ corresponds to $P$ , $B$ to $Q$ and $C$ to $R$ (lesson 8.1).
A map is the ground drawn smaller: every distance is divided by the same number, so the map keeps the shape (chapter 7).

What Similar Means

A photograph and its enlargement, a car and a model of it, a map and the ground it shows: each pair has the same shape but a different size. Figures like these are similar.

In the drawing, every side of $\triangle PQR$ is 1.5 times the corresponding side of $\triangle ABC$ : $4 \times 1.5 = 6$ , $6 \times 1.5 = 9$ and $5 \times 1.5 = 7.5$ . The angles have not changed: the arcs mark the equal pairs.

Similar figures

Two figures are similar if they have the same shape. For two polygons this means both of these are true:

1. all pairs of corresponding angles are equal, and

2. all pairs of corresponding sides are in the same ratio (the sides are proportional).

Scale factor and matching order

" $\triangle ABC$ is similar to $\triangle PQR$ " is written with the corresponding vertices in matching order, as for congruent figures: $A$ with $P$ , $B$ with $Q$ , $C$ with $R$ . Then $\angle A = \angle P$ , $\angle B = \angle Q$ , $\angle C = \angle R$ and

$\frac{PQ}{AB} = \frac{QR}{BC} = \frac{RP}{CA} = k\text{.}$

The number $k$ is the scale factor from $\triangle ABC$ to $\triangle PQR$ : each length of $\triangle PQR$ is $k$ times the corresponding length of $\triangle ABC$ . The new figure goes on top of the fraction. Above, $k = \frac{6}{4} = 1.5$ .

Example 1: Which rectangles are similar?

Rectangle A is 6 cm by 4 cm, rectangle B is 9 cm by 6 cm and rectangle C is 8 cm by 5 cm. Which of B and C is similar to A? Give the scale factor from A where there is one.

Solution

Every angle of a rectangle is $90^{\circ}$ , so the angles always match. Only the sides need checking: long side with long side, short side with short side.

B against A: $\frac{9}{6} = 1.5$ and $\frac{6}{4} = 1.5$ . The ratios are equal, so B is similar to A, with scale factor $1.5$ .

C against A: $\frac{8}{6} = 1.33\ldots$ and $\frac{5}{4} = 1.25$ . The ratios are not equal, so C is not similar to A. (C is too long for its width: it is a different shape.)

B is similar to A, with scale factor $1.5$ ; C is not similar to A.

Why Both Conditions Are Needed

One condition without the other is not enough. Equal angles can come with sides in different ratios, and sides in one ratio can come with different angles.

Example 2: One condition only

(a) Is a rectangle 8 cm by 4 cm similar to a square of side 4 cm? (b) Is a square of side 5 cm similar to a rhombus of side 5 cm with angles $60^{\circ}$ and $120^{\circ}$ ?

Solution

(a) Angles: every angle of both is $90^{\circ}$ , so the angles are equal.

Sides: $\frac{4}{8} = 0.5$ for the long sides and $\frac{4}{4} = 1$ for the short sides. The ratios are different, so the sides are not proportional.

The rectangle is twice as long as it is wide; the square is not. Not similar.

(b) Sides: every side of both is 5 cm, so every ratio is $\frac{5}{5} = 1$ : the sides are proportional.

Angles: the square has $90^{\circ}$ angles and the rhombus has $60^{\circ}$ and $120^{\circ}$ . The angles are not equal. Not similar.

(a) No: equal angles, but the sides are not proportional. (b) No: proportional sides, but the angles are not equal.

Check both, every time

For a polygon, always check both the angles and the ratios of the sides. In Secondary 3 you will learn that for two triangles fewer facts are enough; until then, check both conditions.

Figures That Are Always Similar

Some kinds of figure have only one shape, so any two of them are similar:

All squares are similar: every angle is $90^{\circ}$ , and for squares of sides $a$ and $b$ every pair of sides has the ratio $\frac{b}{a}$ .

All equilateral triangles are similar: every angle is $60^{\circ}$ , and every pair of sides has the same ratio.

All circles are similar: a circle of radius $r$ enlarged by the scale factor $\frac{R}{r}$ is a circle of radius $R$ .

The same is true of regular polygons with the same number of sides, such as any two regular hexagons. But rectangles, rhombuses, isosceles triangles and right-angled triangles come in many shapes, so two of them may or may not be similar.

Congruent figures are similar

Congruent figures have equal corresponding angles, and every pair of corresponding sides has the ratio 1. So congruent figures are similar, with scale factor 1.

Similar figures are congruent only when the scale factor is 1.

Example 3: Always, sometimes or never?

Are these statements true or false? Give a reason. (a) Any two equilateral triangles are similar. (b) Any two rectangles are similar. (c) Any two isosceles triangles are similar.

Solution

(a) True. Every angle of an equilateral triangle is $60^{\circ}$ , so the angles match. If the sides are $a$ and $b$ , all three pairs have the ratio $\frac{b}{a}$ .

(b) False. The rectangle 8 cm by 4 cm and the square of side 4 cm in Example 2 are both rectangles, and they are not similar.

(c) False. One isosceles triangle can have angles $40^{\circ}$ , $70^{\circ}$ and $70^{\circ}$ , and another $100^{\circ}$ , $40^{\circ}$ and $40^{\circ}$ . Their angles are not equal, so they are not similar.

(a) True (b) False (c) False

Deciding Whether Two Figures Are Similar

To decide, pair up the corresponding angles first (equal angles go together), then divide each side of one figure by the corresponding side of the other. When the figure is turned or flipped, pair the sides from the corners, not from where they sit on the page: for example, in Example 4 below, the side between the $66^{\circ}$ and $78^{\circ}$ angles of one triangle matches the side between the $66^{\circ}$ and $78^{\circ}$ angles of the other.

Example 4: Two triangles, one turned over

Show that the two triangles are similar. Write the similarity in matching order, starting with $\triangle ABC$ , and find the scale factor from $\triangle ABC$ to the other triangle.

Solution

Pair the angles: $\angle A = \angle Y = 66^{\circ}$ , $\angle B = \angle Z = 78^{\circ}$ , $\angle C = \angle X = 36^{\circ}$ . So $A$ , $B$ , $C$ correspond to $Y$ , $Z$ , $X$ , and all corresponding angles are equal.

Pair the sides in the same order and divide (new triangle on top):

$\frac{YZ}{AB} = \frac{13.5}{9} = 1.5\text{,}$ $\frac{ZX}{BC} = \frac{21}{14} = 1.5\text{,}$ $\frac{XY}{CA} = \frac{22.5}{15} = 1.5\text{.}$

All three ratios are equal, so the sides are proportional. Both conditions hold.

$\triangle ABC$ is similar to $\triangle YZX$ , with scale factor $1.5$ .

Example 5: Trapeziums with the same angles

Each of the trapeziums $ABCD$ , $PQRS$ and $WXYZ$ has angles $60^{\circ}$ , $60^{\circ}$ , $120^{\circ}$ and $120^{\circ}$ , in that order round the figure. Which of $PQRS$ and $WXYZ$ is similar to $ABCD$ ?

Solution

The angles of all three match in order, so only the sides need checking.

$PQRS$ against $ABCD$ : $\frac{PQ}{AB} = \frac{10}{8} = 1.25$ but $\frac{QR}{BC} = \frac{4}{4} = 1$ . The ratios differ, so $PQRS$ is not similar to $ABCD$ , although its angles are the same.

$WXYZ$ against $ABCD$ : $\frac{WX}{AB} = \frac{12}{8} = 1.5\text{,}$ $\frac{XY}{BC} = \frac{6}{4} = 1.5\text{,}$ $\frac{YZ}{CD} = \frac{6}{4} = 1.5\text{,}$ $\frac{ZW}{DA} = \frac{6}{4} = 1.5\text{.}$ All equal.

$WXYZ$ is similar to $ABCD$ (scale factor $1.5$ ); $PQRS$ is not.

Example 6: Photo frames

Nadia will enlarge a photograph 10 cm by 15 cm so that it fills a frame opening exactly. One frame has an opening 12 cm by 18 cm, and another 13 cm by 18 cm. Which opening can an enlargement of the photograph fill exactly?

Solution

An enlargement is similar to the photograph, so it can fill an opening exactly only if the opening is similar to the photograph. All the angles are right angles: compare short side with short side and long side with long side.

First opening: $\frac{12}{10} = 1.2$ and $\frac{18}{15} = 1.2$ . Equal, so it is similar, with scale factor $1.2$ .

Second opening: $\frac{13}{10} = 1.3$ and $\frac{18}{15} = 1.2$ . Not equal, so it is not similar: enlarged to 13 cm wide, the photograph would be $15 \times 1.3 = 19.5$ cm long, too long for the 18 cm opening.

An enlargement can fill the 12 cm by 18 cm opening exactly (scale factor $1.2$ ); the 13 cm by 18 cm opening is not similar to the photograph.

Common Mistakes

Calling two figures similar because their angles are equal. A rectangle 8 cm by 4 cm and a square have equal angles and are not similar.
Calling two figures similar because their sides are in one ratio. A square and a rhombus with the same side are not similar.
Adding instead of multiplying. Adding 4 cm to every side of a 10 cm by 6 cm rectangle gives 14 cm by 10 cm, which is a different shape.
Pairing the sides by their places on the page. Pair them from the corresponding corners: equal angles first, then the sides between them.
Writing the scale factor upside down. From $\triangle ABC$ to $\triangle PQR$ it is $\frac{PQ}{AB}$ : the figure you go to is on top.

Practice Questions

Try each question before you open its solution. Check both conditions, and say which ratio you worked out.

Question 1

Is a rectangle 5 cm by 3 cm similar to a rectangle 15 cm by 9 cm? If so, give the scale factor from the smaller to the larger.

ShowHide solution

All angles are $90^{\circ}$ . $\frac{15}{5} = 3$ and $\frac{9}{3} = 3$ : equal ratios.

Yes, with scale factor $3$ .

Question 2

Which of these rectangles are similar to a rectangle 12 cm by 8 cm? (a) 18 cm by 12 cm (b) 15 cm by 10 cm (c) 16 cm by 12 cm

ShowHide solution

(a) $\frac{18}{12} = 1.5$ and $\frac{12}{8} = 1.5$ : similar.

(b) $\frac{15}{12} = 1.25$ and $\frac{10}{8} = 1.25$ : similar.

(a) and (b) are similar to it; (c) is not.

Question 3

True or false? Give a reason. (a) Any two circles are similar. (b) Any two right-angled triangles are similar. (c) Any two regular hexagons are similar. (d) Two congruent triangles are similar.

ShowHide solution

(a) True: one circle is an enlargement of the other, by the ratio of their radii.

(b) False: one right-angled triangle can have angles $90^{\circ}$ , $45^{\circ}$ , $45^{\circ}$ and another $90^{\circ}$ , $30^{\circ}$ , $60^{\circ}$ . The angles are not equal.

(d) True: corresponding angles are equal and every pair of sides has the ratio 1.

(a) True (b) False (c) True (d) True

Question 4

$\triangle ABC$ is similar to $\triangle PQR$ . (a) Write the three equal ratios of sides. (b) $\angle A = 50^{\circ}$ and $\angle Q = 60^{\circ}$ . Find $\angle B$ , $\angle P$ and $\angle R$ .

ShowHide solution

(a) $A$ , $B$ , $C$ correspond to $P$ , $Q$ , $R$ : $\frac{PQ}{AB} = \frac{QR}{BC} = \frac{RP}{CA}\text{.}$

(b) Corresponding angles of similar figures are equal: $\angle B = \angle Q = 60^{\circ}$ and $\angle P = \angle A = 50^{\circ}$ .

$\angle R = 180^{\circ} - 50^{\circ} - 60^{\circ} = 70^{\circ}$ (∠ sum of △).

(a) $\frac{PQ}{AB} = \frac{QR}{BC} = \frac{RP}{CA}$ (b) $\angle B = 60^{\circ}$ , $\angle P = 50^{\circ}$ , $\angle R = 70^{\circ}$

Question 5

The two triangles are similar, and the arcs mark the equal angles. Write the similarity in matching order, starting with $\triangle LMN$ .

ShowHide solution

One arc: $L$ and $U$ . Two arcs: $M$ and $S$ . Three arcs: $N$ and $T$ .

So $L$ , $M$ , $N$ correspond to $U$ , $S$ , $T$ .

$\triangle LMN$ is similar to $\triangle UST$ .

Question 6

In $\triangle ABC$ , $\angle A = 40^{\circ}$ , $\angle B = 60^{\circ}$ and $\angle C = 80^{\circ}$ . In $\triangle DEF$ , $\angle D = 60^{\circ}$ , $\angle E = 80^{\circ}$ and $\angle F = 40^{\circ}$ . The sides are shown. Show that the triangles are similar, write the similarity starting with $\triangle ABC$ , and find the scale factor from $\triangle ABC$ to $\triangle DEF$ .

ShowHide solution

Pair the angles: $\angle A = \angle F = 40^{\circ}$ , $\angle B = \angle D = 60^{\circ}$ , $\angle C = \angle E = 80^{\circ}$ . So $A$ , $B$ , $C$ correspond to $F$ , $D$ , $E$ .

Sides in the same order: $\frac{FD}{AB} = \frac{9.8}{19.6} = 0.5\text{,}$ $\frac{DE}{BC} = \frac{6.4}{12.8} = 0.5\text{,}$ $\frac{EF}{CA} = \frac{8.6}{17.2} = 0.5\text{.}$ All equal.

$\triangle ABC$ is similar to $\triangle FDE$ , with scale factor $0.5$ .

Question 7

Rhombus P has sides 6 cm and angles $70^{\circ}$ and $110^{\circ}$ . Rhombus Q has sides 9 cm and angles $70^{\circ}$ and $110^{\circ}$ . Rhombus R has sides 9 cm and angles $60^{\circ}$ and $120^{\circ}$ . Which of Q and R is similar to P?

ShowHide solution

Q: every side ratio is $\frac{9}{6} = 1.5$ , and the angles are the same as P's. Similar, with scale factor $1.5$ .

R: every side ratio is $\frac{9}{6} = 1.5$ , but its angles ( $60^{\circ}$ and $120^{\circ}$ ) are not P's. Not similar.

Q is similar to P; R is not.

Question 8

A rectangle 4 cm by 6 cm is drawn again, larger, so that the new rectangle is similar to it and its shorter side is 10 cm. Find the scale factor and the longer side of the new rectangle.

ShowHide solution

Scale factor $= \frac{\text{new shorter side}}{\text{old shorter side}} = \frac{10}{4} = 2.5\text{.}$

Longer side $= 6 \times 2.5 = 15$ cm. Check: $\frac{15}{6} = 2.5$ .

Scale factor $2.5$ ; the longer side is $15$ cm.

Question 9

Wei Jie says that a rectangle 8 cm by 5 cm and a rectangle 11 cm by 8 cm are similar, because each side of the second is 3 cm longer. Is he right?

ShowHide solution

Similar figures have their sides multiplied by the same number, not increased by the same amount.

$\frac{11}{8} = 1.375$ and $\frac{8}{5} = 1.6$ . The ratios are not equal.

No. The ratios $1.375$ and $1.6$ differ, so the rectangles are not similar.

Question 10

The flag of Singapore has its width and length in the ratio $2 : 3$ . A school has a flag 60 cm by 90 cm, a large flag 120 cm by 180 cm and a banner 100 cm by 160 cm. Which of the large flag and the banner is similar to the 60 cm by 90 cm flag?

ShowHide solution

Large flag: $\frac{120}{60} = 2$ and $\frac{180}{90} = 2$ . Similar, scale factor 2.

Banner: $\frac{100}{60} = 1.66\ldots$ and $\frac{160}{90} = 1.77\ldots$ . Not similar. (Its width and length are in the ratio $100 : 160 = 5 : 8$ , not $2 : 3$ .)

The large flag is similar to it (scale factor 2); the banner is not.

Question 11

A television screen is 80 cm wide and 45 cm high. A second screen is 120 cm wide and 67.5 cm high, and a phone screen is 15 cm by 7 cm. Which of the other two screens is similar to the first?

ShowHide solution

Second screen: $\frac{120}{80} = 1.5$ and $\frac{67.5}{45} = 1.5$ . Similar, scale factor $1.5$ .

Phone: $\frac{15}{80} = 0.1875$ and $\frac{7}{45} = 0.155\ldots$ . Not similar.

The 120 cm by 67.5 cm screen is similar to the first; the phone screen is not.

Question 12

Trapezium $ABCD$ has $AB = 8$ cm, $BC = 4$ cm, $CD = 4$ cm, $DA = 4$ cm and angles $60^{\circ}$ , $60^{\circ}$ , $120^{\circ}$ , $120^{\circ}$ . A trapezium $EFGH$ similar to it has $EF = 6$ cm. Find the scale factor from $ABCD$ to $EFGH$ , and the other three sides of $EFGH$ .

ShowHide solution

$EF$ corresponds to $AB$ : scale factor $= \frac{EF}{AB} = \frac{6}{8} = 0.75\text{.}$

$FG = 4 \times 0.75 = 3$ cm, $GH = 4 \times 0.75 = 3$ cm, $HE = 4 \times 0.75 = 3$ cm.

The scale factor is less than 1, so $EFGH$ is smaller than $ABCD$ , as $6 < 8$ shows.

Scale factor $0.75$ ; $FG = GH = HE = 3$ cm.

Key Points

Similar figures have the same shape, but not necessarily the same size.
Two polygons are similar only if corresponding angles are equal and corresponding sides are in the same ratio.
Equal angles alone, or sides in one ratio alone, are not enough: a rectangle and a square, a square and a rhombus.
The scale factor from figure A to figure B is a length of B divided by the corresponding length of A.
Name similar figures in matching order: " $\triangle ABC$ is similar to $\triangle PQR$ " means $\angle A = \angle P$ , $\angle B = \angle Q$ , $\angle C = \angle R$ and $\frac{PQ}{AB} = \frac{QR}{BC} = \frac{RP}{CA}\text{.}$
All circles, all squares and all equilateral triangles are similar.
Congruent figures are similar, with scale factor 1.