Sec 2 Maths · Congruence and similarity

Similarity tests questions with answers

Practice questions on similarity tests, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Triangles $ABC$ and $PQR$ are similar. Using the parts marked, state the similarity test: AA, SSS or SAS.

Step 1 of 2

Compare the marked parts

\mathstrut \dfrac{PR}{AB} = \dfrac{15}{10}

\mathstrut \dfrac{RQ}{BC} = \dfrac{7.5}{5}

\mathstrut \dfrac{QP}{CA} = \dfrac{12}{8}

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Questions

Question 1Name the similarity test
Triangles $ABC$ and $PQR$ are similar. Using the parts marked, state the similarity test: AA, SSS or SAS.
Show the worked solution
1. Step 1: Compare the marked parts $\mathstrut \dfrac{QR}{AB} = \dfrac{15}{10}$ $\mathstrut \dfrac{PQ}{CA} = \dfrac{13.5}{9}$ $\angle BAC = \angle RQP$
2. Step 2: Name the testTwo pairs of sides are in the same ratio, $\dfrac{3}{2}$ , and the angles between them are equal: SAS.
Question 2Find a side of similar triangles
In the diagram, triangle $ABC$ is similar to triangle $PRQ$ . Find $x$ .
Show the worked solution
1. Step 1: Match the sides by the order of the letters $CA$ matches $QP$ and $AB$ matches $PR$ .
2. Step 2: Matching sides are in the same ratio $\dfrac{AB}{PR} = \dfrac{CA}{QP}$ $\mathstrut \dfrac{x}{18} = \dfrac{9}{13.5}$
3. Step 3: Solve for x $x = \dfrac{9}{13.5} \times 18$ $x = 12$ cm
Question 3A line parallel to a side
In the diagram, $D$ lies on $AB$ and $E$ lies on $AC$ , with $DE$ parallel to $BC$ . Find $DE$ .
Show the worked solution
1. Step 1: DE ∥ BC, so triangle ADE is similar to triangle ABC $\angle ADE = \angle ABC$ (corresponding angles) $\angle A$ is common, so the triangles are similar (AA).
2. Step 2: Use the whole side AB, not DB $\mathstrut AB = 4 + 6$ $AB = 10$
3. Step 3: Matching sides are in the same ratio $\dfrac{DE}{BC} = \dfrac{AD}{AB}$ $\mathstrut \dfrac{DE}{15} = \dfrac{4}{10}$ $DE = \dfrac{4}{10} \times 15$ $DE = 6$ cm
Question 4Crossing lines with parallel sides
In the diagram, the lines $AD$ and $BC$ meet at $X$ , and $AB$ is parallel to $DC$ . Find $XC$ , correct to 3 significant figures.
Show the worked solution
1. Step 1: AB ∥ DC, so triangle XAB is similar to triangle XDC $\angle XAB = \angle XDC$ (alternate angles) $\angle AXB = \angle DXC$ (vertically opposite angles)
2. Step 2: Matching sides are in the same ratio $\dfrac{XC}{XB} = \dfrac{DC}{AB}$ $\mathstrut \dfrac{XC}{5} = \dfrac{10}{6}$ $XC = \dfrac{10}{6} \times 5$ $XC = 8.3333\ldots$ $\phantom{XC} = 8.33$ cm (3 s.f.)

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Worked example

Triangles

ABC

and

PQR

are similar. Using the parts marked, state the similarity test: AA, SSS or SAS.

Step 1 of 2

Compare the marked parts

\mathstrut \dfrac{PR}{AB} = \dfrac{15}{10}

\mathstrut \dfrac{RQ}{BC} = \dfrac{7.5}{5}

\mathstrut \dfrac{QP}{CA} = \dfrac{12}{8}

Questions

Question 1Name the similarity test

Triangles $ABC$ and $PQR$ are similar. Using the parts marked, state the similarity test: AA, SSS or SAS.

Show the worked solution

Step 1: Compare the marked parts $\mathstrut \dfrac{QR}{AB} = \dfrac{15}{10}$ $\mathstrut \dfrac{PQ}{CA} = \dfrac{13.5}{9}$ $\angle BAC = \angle RQP$
Step 2: Name the testTwo pairs of sides are in the same ratio, $\dfrac{3}{2}$ , and the angles between them are equal: SAS.

Question 2Find a side of similar triangles

In the diagram, triangle $ABC$ is similar to triangle $PRQ$ . Find $x$ .

Show the worked solution

Step 1: Match the sides by the order of the letters $CA$ matches $QP$ and $AB$ matches $PR$ .
Step 2: Matching sides are in the same ratio $\dfrac{AB}{PR} = \dfrac{CA}{QP}$ $\mathstrut \dfrac{x}{18} = \dfrac{9}{13.5}$
Step 3: Solve for x $x = \dfrac{9}{13.5} \times 18$ $x = 12$ cm

Question 3A line parallel to a side

In the diagram, $D$ lies on $AB$ and $E$ lies on $AC$ , with $DE$ parallel to $BC$ . Find $DE$ .

Show the worked solution

Step 1: DE ∥ BC, so triangle ADE is similar to triangle ABC $\angle ADE = \angle ABC$ (corresponding angles) $\angle A$ is common, so the triangles are similar (AA).
Step 2: Use the whole side AB, not DB $\mathstrut AB = 4 + 6$ $AB = 10$
Step 3: Matching sides are in the same ratio $\dfrac{DE}{BC} = \dfrac{AD}{AB}$ $\mathstrut \dfrac{DE}{15} = \dfrac{4}{10}$ $DE = \dfrac{4}{10} \times 15$ $DE = 6$ cm

Question 4Crossing lines with parallel sides

In the diagram, the lines $AD$ and $BC$ meet at $X$ , and $AB$ is parallel to $DC$ . Find $XC$ , correct to 3 significant figures.

Show the worked solution

Step 1: AB ∥ DC, so triangle XAB is similar to triangle XDC $\angle XAB = \angle XDC$ (alternate angles) $\angle AXB = \angle DXC$ (vertically opposite angles)
Step 2: Matching sides are in the same ratio $\dfrac{XC}{XB} = \dfrac{DC}{AB}$ $\mathstrut \dfrac{XC}{5} = \dfrac{10}{6}$ $XC = \dfrac{10}{6} \times 5$ $XC = 8.3333\ldots$ $\phantom{XC} = 8.33$ cm (3 s.f.)