Year 9 Similar Triangles Worksheets with Answers

Worked examples

Year 9 similar triangles: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Testing whether two triangles are similar (SSS)

   Are these pairs of triangles similar? Give a reason.

   (a) Triangle $A$ has sides $5$ cm, $8$ cm and $10$ cm. Triangle $B$ has sides $7.5$ cm, $12$ cm and $15$ cm.

   (b) Triangle $C$ has sides $4$ cm, $6$ cm and $7$ cm. Triangle $D$ has sides $8$ cm, $12$ cm and $15$ cm.

   Show worked solution

   Key idea: Two triangles are similar (SSS) if all three pairs of matching sides are in the same ratio. Match shortest with shortest, middle with middle and longest with longest, then divide each larger side by its smaller partner.

   1. (a) Divide each side of triangle $B$ by the matching side of triangle $A$:
      $\frac{7.5}{5} = 1.5 \frac{12}{8} = 1.5 \frac{15}{10} = 1.5$
   2. All three ratios are equal, so the triangles are similar, with scale factor $1.5$.
   3. (b) Divide each side of triangle $D$ by the matching side of triangle $C$:
      $\frac{8}{4} = 2 \frac{12}{6} = 2 \frac{15}{7} \approx 2.14$
   4. The third ratio is different, so the triangles are not similar. Two matching ratios aren't enough: all three must agree.

   Answer: (a) Similar (SSS, scale factor $1.5$). (b) Not similar, because $\dfrac{15}{7} \neq 2$.

2. Example 2 · Finding missing sides with the scale factor

   $\triangle ABC \sim \triangle PQR$, with $A$ matching $P$, $B$ matching $Q$ and $C$ matching $R$. $AB = 6$ cm, $BC = 8$ cm, $AC = 10$ cm, $PQ = 9$ cm and $QR = 12$ cm.

   (a) Find the scale factor from $\triangle ABC$ to $\triangle PQR$.

   (b) Find $PR$.

   (c) A third similar triangle, $\triangle XYZ$, has $XZ = 4$ cm matching $AC$. Find $XY$.

   Show worked solution

   Key idea: The order of the letters tells you which sides match: $AB \leftrightarrow PQ$, $BC \leftrightarrow QR$, $AC \leftrightarrow PR$. The scale factor is $\dfrac{\text{image side}}{\text{original side}}$ for any matching pair.

   1. (a) Use the pair where both lengths are known, $AB$ and $PQ$:
      $k = \frac{PQ}{AB} = \frac{9}{6} = 1.5$ Check with the second pair: $\dfrac{QR}{BC} = \dfrac{12}{8} = 1.5$ ✓
   2. (b) $PR$ matches $AC$. Multiply by the scale factor:
      $PR = 1.5 \times 10 = 15 \text{ cm}$
   3. (c) From $\triangle ABC$ to $\triangle XYZ$ the scale factor is
      $k = \frac{XZ}{AC} = \frac{4}{10} = 0.4$ A scale factor less than $1$ means the triangle is smaller.
   4. $XY$ matches $AB$:
      $XY = 0.4 \times 6 = 2.4 \text{ cm}$

   Answer: (a) $1.5$, (b) $PR = 15$ cm, (c) $XY = 2.4$ cm

3. Example 3 · A triangle inside a triangle (parallel line)

   In $\triangle ABC$, point $D$ is on $AB$ and point $E$ is on $AC$, with $DE \parallel BC$. $AD = 4$ cm, $DB = 6$ cm and $DE = 5$ cm. Find $BC$.

   Show worked solution

   Key idea: A line parallel to one side cuts off a smaller triangle that is similar to the whole triangle. Compare the whole side $AB$ with $AD$, not the piece $DB$.

   1. Show the triangles are similar (AAA). $\angle A$ is shared. $\angle ADE = \angle ABC$ because they are corresponding angles on parallel lines. So $\triangle ADE \sim \triangle ABC$.
   2. Find the whole side:
      $AB = AD + DB = 4 + 6 = 10 \text{ cm}$
   3. Scale factor from the small triangle to the large one:
      $k = \frac{AB}{AD} = \frac{10}{4} = 2.5$
   4. $BC$ matches $DE$:
      $BC = 2.5 \times 5 = 12.5 \text{ cm}$
   5. Check with a ratio: $\dfrac{BC}{DE} = \dfrac{12.5}{5} = 2.5 = \dfrac{AB}{AD}$ ✓

   Answer: $BC = 12.5$ cm

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