Problems Involving Congruence and Similarity

Decide whether two figures are congruent, similar or neither, use the scale factor for perimeters, and solve problems with ramps, shadows, photo frames and A-size paper, giving a reason at each step.

Before you start

Congruent figures: the same shape and size; corresponding sides and angles are equal (lesson 8.1).
Similar figures: corresponding angles equal and corresponding sides in the same ratio, the scale factor $k$ ; an unknown side comes from two equal ratios (lessons 8.2 and 8.3).
An image with scale factor $k$ has every length $k$ times the object's, and the same angles; so the perimeter of the image is $k$ times the perimeter of the object (lesson 8.4).

Congruent, Similar or Neither?

Given every side and angle of two figures, three answers are possible, and one check decides between them.

Deciding

1. Pair up the corresponding angles. If they are not all equal, the figures are neither congruent nor similar.

2. If they are all equal, divide each side of the second figure by the corresponding side of the first.

All the ratios equal to 1: congruent (and similar, with $k = 1$ ). All the ratios equal to the same $k \neq 1$ : similar, not congruent. Ratios that differ: neither.

Example 1: Four pairs

Is each pair congruent, similar (give the scale factor from the first to the second) or neither? (a) A rectangle 6 cm by 10 cm and a rectangle 10 cm by 6 cm. (b) A rectangle 6 cm by 10 cm and a rectangle 9 cm by 15 cm. (c) A rectangle 6 cm by 10 cm and a rectangle 8 cm by 12 cm. (d) A triangle with sides 8 cm, 10 cm, 12 cm and angles $41^{\circ}$ , $56^{\circ}$ , $83^{\circ}$ , and a triangle with sides 6 cm, 7.5 cm, 9 cm and the same angles, each side opposite the same angle as in the first.

Solution

The angles of every rectangle are $90^{\circ}$ , and in (d) the angles are the same. So in every pair the angles match, and the ratios decide.

(a) The second is the first turned: $\frac{6}{6} = 1$ and $\frac{10}{10} = 1$ . Congruent.

(b) $\frac{9}{6} = 1.5$ and $\frac{15}{10} = 1.5$ . Similar, with $k = 1.5$ .

(d) $\frac{6}{8} = \frac{7.5}{10} = \frac{9}{12} = 0.75\text{.}$ Similar, with $k = 0.75$ .

(a) congruent (b) similar, $k = 1.5$ (c) neither (d) similar, $k = 0.75$

Problems in Real Settings

In a longer problem, say at each step which figures are congruent or similar and why you may use it (given, symmetric, or one is an enlargement of the other), then which sides or angles correspond.

Example 2: A-size paper

A sheet of A4 paper is 210 mm by 297 mm. Folded in half across its longer side, it makes a sheet 148.5 mm by 210 mm, the size of A5 (A5 is sold as 148 mm by 210 mm). (a) Show that the ratios of corresponding sides of this half-sheet and A4 agree to 3 decimal places. (b) Find the scale factor from A4 to the half-sheet. (c) Find the perimeters of A4 and the half-sheet, and check that their ratio is the scale factor.

Solution

(a) Both are rectangles, so the angles match. Compare longer side with longer side and shorter with shorter: $\frac{210}{297} = 0.70707\ldots$ and $\frac{148.5}{210} = 0.70714\ldots\text{,}$ both $0.707$ to 3 decimal places. (The millimetre sizes of A-size paper are rounded, which is why the ratios differ in the fourth decimal place.) A-size paper is designed so that each half is exactly similar to the whole, so A5 is similar to A4.

(b) $k = 0.707$ (to 3 decimal places), a reduction.

$\frac{717}{1014} = 0.707$ (to 3 decimal places), the scale factor.

(a) $\frac{210}{297}$ and $\frac{148.5}{210}$ are both $0.707$ to 3 decimal places (b) $k = 0.707$ (c) $1014$ mm and $717$ mm

Example 3: A loading ramp

A straight loading ramp $AC$ rises from $A$ on the ground to $C$ , 0.9 m above the point $B$ , and $AB = 6$ m. Upright supports $DE$ and $FG$ stand on $AB$ , 2 m and 4 m from $A$ , under the ramp. $\triangle ADE$ and $\triangle AFG$ are each similar to $\triangle ABC$ . Find the heights of the two supports.

Solution

$A$ , $D$ , $E$ correspond to $A$ , $B$ , $C$ , and $AB = 2 + 2 + 2 = 6$ m.

$\frac{DE}{BC} = \frac{AD}{AB}$ , so $DE = 0.9 \times \frac{2}{6} = 0.3$ m.

In the same way $\frac{FG}{BC} = \frac{AF}{AB}$ , so $FG = 0.9 \times \frac{4}{6} = 0.6$ m.

Check: $F$ is twice as far from $A$ as $D$ , and $FG$ is twice $DE$ .

$DE = 0.3$ m and $FG = 0.6$ m.

Example 4: A photo frame

A photograph 12 cm by 18 cm is put in a frame with a border 3 cm wide all round. (a) Is the outside edge of the frame similar to the photograph? (b) The frame maker keeps the border 3 cm wide along the long sides, so the outside is 18 cm wide. How long must the outside be for it to be similar to the photograph, and how wide is the border at each end?

Solution

(a) The outside is $12 + 3 + 3 = 18$ cm by $18 + 3 + 3 = 24$ cm. $\frac{18}{12} = 1.5$ but $\frac{24}{18} = 1.33\ldots$ . Not similar: a border of one width all round makes a rectangle of a different shape.

(b) For the outside to be similar, $k = \frac{18}{12} = 1.5$ , so its length is $1.5 \times 18 = 27$ cm.

The photograph is 18 cm long, so the ends of the border take $27 - 18 = 9$ cm, which is $9 \div 2 = 4.5$ cm at each end.

(a) No ( $18$ cm by $24$ cm) (b) $27$ cm long, with $4.5$ cm of border at each end

Example 5: Shadows at two times

At 9 a.m. a pole 2 m tall casts a shadow 3 m long, and a tree casts a shadow 12 m long. At each time, the triangles formed by the pole and the tree with their shadows are similar. (a) Find the height of the tree. (b) Later the pole's shadow is 1.5 m long. How long is the tree's shadow then?

Solution

(a) At 9 a.m. the two triangles are similar (given): $\frac{\text{tree}}{2} = \frac{12}{3}$ , so the tree is $2 \times 4 = 8$ m tall.

(b) Later the sun is higher, so the 9 a.m. ratio of shadow to height ( $3 : 2$ ) no longer holds. At the new time the triangles are again similar (given): $\frac{\text{tree's shadow}}{1.5} = \frac{8}{2}\text{,}$ so the shadow is $1.5 \times 4 = 6$ m.

(a) $8$ m (b) $6$ m

Example 6: Congruence and similarity together

A logo is the triangle $ABC$ , symmetric about the line $AM$ , where $M$ is on $BC$ and $AM$ is perpendicular to $BC$ . So $\triangle ABM \equiv \triangle ACM$ . $P$ is on $AB$ and $Q$ on $AM$ , with $PQ \parallel BM$ , and $\triangle APQ$ is similar to $\triangle ABM$ . $AP = 4$ cm, $PB = 6$ cm and $BM = 6$ cm. Find (a) $BC$ , (b) the perimeter of $\triangle ABC$ , (c) $PQ$ .

Solution

(a) $\triangle ABM \equiv \triangle ACM$ , so $MC = MB = 6$ cm (corr. sides of congruent △s). $BC = 6 + 6 = 12$ cm.

(b) $AB = 4 + 6 = 10$ cm, and $AC = AB = 10$ cm (corr. sides of congruent △s). Perimeter $= 10 + 10 + 12 = 32$ cm.

(c) $A$ , $P$ , $Q$ correspond to $A$ , $B$ , $M$ . $\frac{PQ}{BM} = \frac{AP}{AB}$ , so $PQ = 6 \times \frac{4}{10} = 2.4$ cm.

(a) $12$ cm (b) $32$ cm (c) $2.4$ cm

Common Mistakes

Calling two figures similar from their angles alone, or from one ratio of sides. Check all the angles and all the ratios.
Forgetting that congruent figures are similar too, with $k = 1$ .
Adding the same amount to every side (a border of one width) and expecting a similar figure.
Using the ratio of shadow to height from one time of day at another: shadows change with the sun.
Giving no reason. Say which figures are congruent or similar, and which parts correspond, at each step.

Practice Questions

Try each question before you open its solution. Give a reason at each step.

Question 1

Congruent, similar or neither? (a) A square of side 4 cm and a square of side 6 cm. (b) A rectangle 5 cm by 8 cm and a rectangle 8 cm by 5 cm. (c) A rectangle 4 cm by 6 cm and a rectangle 6 cm by 8 cm. (d) Two circles, each of radius 2 cm.

ShowHide solution

(a) All squares are similar: $k = \frac{6}{4} = 1.5$ . Similar, not congruent.

(b) The same rectangle turned: congruent.

(d) The same radius: congruent.

(a) similar (b) congruent (c) neither (d) congruent

Question 2

$\triangle ABC$ has $AB = 8$ cm, $BC = 5$ cm, $CA = 7$ cm, $\angle A = 38^{\circ}$ , $\angle B = 60^{\circ}$ , $\angle C = 82^{\circ}$ . $\triangle XYZ$ has $XY = 10$ cm, $YZ = 14$ cm, $ZX = 16$ cm, $\angle X = 60^{\circ}$ , $\angle Y = 82^{\circ}$ , $\angle Z = 38^{\circ}$ . Are they congruent, similar or neither? If similar, write the similarity starting with $\triangle ABC$ and give the scale factor.

ShowHide solution

Angles: $\angle A = \angle Z = 38^{\circ}$ , $\angle B = \angle X = 60^{\circ}$ , $\angle C = \angle Y = 82^{\circ}$ . So $A$ , $B$ , $C$ correspond to $Z$ , $X$ , $Y$ .

Ratios: $\frac{ZX}{AB} = \frac{16}{8} = 2$ , $\frac{XY}{BC} = \frac{10}{5} = 2$ , $\frac{YZ}{CA} = \frac{14}{7} = 2$ . All equal, and not 1.

Similar: $\triangle ABC$ is similar to $\triangle ZXY$ , with $k = 2$ .

Question 3

A rectangle 4 cm by 6 cm is similar to a rectangle whose longer side is 15 cm. Find the perimeter of each.

ShowHide solution

First: $2 \times (4 + 6) = 20$ cm. $k = \frac{15}{6} = 2.5$ .

Second: $2.5 \times 20 = 50$ cm. Check: its shorter side is $2.5 \times 4 = 10$ cm, and $2 \times (10 + 15) = 50$ cm.

$20$ cm and $50$ cm

Question 4

A straight ramp rises 1.2 m over a level distance of 8 m. An upright post stands on the ground under the ramp, 5 m from its lower end, and reaches up to the ramp. The small triangle at the lower end is similar to the whole triangle. Find the height of the post.

ShowHide solution

The post matches the 1.2 m rise, and its 5 m distance matches the 8 m distance.

$\frac{\text{height}}{1.2} = \frac{5}{8}\text{,}$ so the height is $1.2 \times \frac{5}{8} = 0.75$ m.

$0.75$ m

Question 5

At 9 a.m. Mei Ling, who is 1.5 m tall, casts a shadow 2.5 m long on level ground, and a lamp post casts a shadow 10 m long. At each time, the triangles formed by Mei Ling and the lamp post with their shadows are similar. (a) Find the height of the lamp post. (b) At noon Mei Ling's shadow is 0.3 m long. How long is the lamp post's shadow then?

ShowHide solution

(a) At 9 a.m. the triangles are similar (given): $\frac{\text{lamp post}}{1.5} = \frac{10}{2.5}\text{,}$ so the lamp post is $1.5 \times 4 = 6$ m tall.

(b) At noon the 9 a.m. ratio of shadow to height no longer holds; the noon triangles are similar (given): $\frac{\text{lamp post's shadow}}{0.3} = \frac{6}{1.5}\text{,}$ so the shadow is $0.3 \times 4 = 1.2$ m.

(a) $6$ m (b) $1.2$ m

Question 6

A photograph 6 cm by 9 cm has a border 1 cm wide all round. (a) Is the outside edge similar to the photograph? (b) The border stays 1 cm wide along the long sides. How wide must it be at each end for the outside to be similar to the photograph?

ShowHide solution

(a) Outside: $6 + 2 = 8$ cm by $9 + 2 = 11$ cm. $\frac{8}{6} = 1.333\ldots$ and $\frac{11}{9} = 1.222\ldots$ . Not equal.

(b) The outside is $6 + 1 + 1 = 8$ cm wide, so for it to be similar, $k = \frac{8}{6} = \frac{4}{3}$ and its length is $\frac{4}{3} \times 9 = 12$ cm.

The ends of the border take $12 - 9 = 3$ cm, which is $3 \div 2 = 1.5$ cm at each end.

(a) No: $8$ cm by $11$ cm is not similar to $6$ cm by $9$ cm. (b) $1.5$ cm at each end

Question 7

The half-sheet of Example 2, 148.5 mm by 210 mm, is folded in half across its longer side to make a sheet 105 mm by 148.5 mm, the size of A6 (sold as 105 mm by 148 mm). (a) Show that this sheet is similar to A4 (210 mm by 297 mm), and find the scale factor from A4 to it. (b) Find its perimeter from the perimeter of A4, 1014 mm.

ShowHide solution

(a) Shorter sides: $\frac{105}{210} = 0.5$ . Longer sides: $\frac{148.5}{297} = 0.5$ . Equal, and all angles are $90^{\circ}$ : similar, with $k = 0.5$ .

(b) $0.5 \times 1014 = 507$ mm. Check: $2 \times (105 + 148.5) = 507$ mm.

(a) $k = 0.5$ (b) $507$ mm

Question 8

A rectangular garden 12 m by 8 m is cut by a straight path along one diagonal into two triangles, which are congruent. The diagonal is 14.4 m long (to 1 decimal place). Mr Lim puts a fence all round each triangle. How much fence does each triangle need? Give a reason.

ShowHide solution

One triangle has sides 12 m, 8 m and 14.4 m: it needs $12 + 8 + 14.4 = 34.4$ m.

The two triangles are congruent, so the other has the same three sides (corr. sides of congruent △s) and needs the same, $34.4$ m.

$34.4$ m for each triangle

Question 9

Triangle $ABC$ is symmetric about $AM$ , where $M$ is on $BC$ and $AM \perp BC$ , so $\triangle ABM \equiv \triangle ACM$ . $P$ is on $AB$ and $Q$ on $AM$ with $PQ \parallel BM$ , and $\triangle APQ$ is similar to $\triangle ABM$ . $AP = 6.5$ cm, $PB = 6.5$ cm and $BM = 5$ cm. Find $BC$ , the perimeter of $\triangle ABC$ and $PQ$ .

ShowHide solution

$MC = MB = 5$ cm (corr. sides of congruent △s), so $BC = 10$ cm.

$AB = 6.5 + 6.5 = 13$ cm and $AC = AB = 13$ cm (corr. sides of congruent △s). Perimeter $= 13 + 13 + 10 = 36$ cm.

$\frac{PQ}{BM} = \frac{AP}{AB}$ , so $PQ = 5 \times \frac{6.5}{13} = 2.5$ cm.

$BC = 10$ cm, perimeter $36$ cm, $PQ = 2.5$ cm

Question 10

A model of a rectangular swimming pool is made to the scale $1 : 250$ . The model is 16 cm long and 8 cm wide. (a) Find the length and width of the real pool, in metres. (b) Find the perimeter of the model and of the real pool, and check that their ratio is the scale.

ShowHide solution

(a) Length $= 16 \times 250 = 4000$ cm $= 40$ m. Width $= 8 \times 250 = 2000$ cm $= 20$ m.

(b) Model: $2 \times (16 + 8) = 48$ cm. Pool: $2 \times (40 + 20) = 120$ m $= 12\,000$ cm.

$\frac{48}{12\,000} = \frac{1}{250}\text{,}$ the scale factor from the pool to the model.

(a) $40$ m by $20$ m (b) $48$ cm and $120$ m

Key Points

Compare all angles first, then all ratios of corresponding sides: all 1, congruent; all the same $k$ , similar; different, neither.
Congruent figures are similar with $k = 1$ ; similar figures are congruent only when $k = 1$ .
A4, A5 and A6 paper are similar: each fold in half gives a sheet of the same shape.
A border of one width all round does not give a similar rectangle.
In a problem with several steps, say at each step which figures are congruent or similar and which parts correspond.