Grade 9

Similar figures: practice problems

Similar figures have the same shape but different sizes. Corresponding sides are in the same ratio.

Basic, Standard, Advanced: Grade 9 · Term 3

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Conditions and ratios

Similar triangles

All three pairs of sides in the same ratio
Two pairs of sides in the same ratio with equal included angles
Two pairs of equal angles

Write $\triangle \mathrm{ABC} \sim \triangle \mathrm{DEF}$ with corresponding vertices in order.

Parallel lines and ratios

If $\mathrm{DE} \parallel \mathrm{BC}$ in $\triangle \mathrm{ABC}$ , then $\mathrm{AD} : \mathrm{AB} = \mathrm{DE} : \mathrm{BC}$ and $\mathrm{AD} : \mathrm{DB} = \mathrm{AE} : \mathrm{EC}$ . To find DE, use AD : AB, not AD : DB.

Midpoint theorem: the segment joining the midpoints of two sides is parallel to the third side and half as long.

Area and volume ratios

Ratios

If the ratio of similarity is $m : n$ :

\text{area ratio} = m^2 : n^2,\qquad \text{volume ratio} = m^3 : n^3

Q1BasicGr 9 · Term 3Similar figures

In the figure, $\triangle \mathrm{ABC} \sim \triangle \mathrm{DEF}$ . Find $x$ .

Answer

x = 25

Solution

The ratio of similarity is
$\mathrm{AB} : \mathrm{DE} = 8 : 20 = 2 : 5$
CA : FD has the same ratio:
$10 : x = 2 : 5 \quad\Rightarrow\quad x = 25$

Q2StandardGr 9 · Term 3Similar figures

In $\triangle \mathrm{ABC}$ , M and N are the midpoints of AB and AC. Find MN.

Answer

8\text{ cm}

Solution

By the midpoint theorem,
$\mathrm{MN} = \frac{1}{2} \times 16 = 8$

Q3AdvancedGr 9 · Term 3Similar figures

In the figure, AB, EF and CD are all perpendicular to BD, and E is the intersection of AD and BC. Find EF.

Answer

4\text{ cm}

Solution

AB ∥ CD gives $\triangle \mathrm{EAB} \sim \triangle \mathrm{EDC}$ , so
$\mathrm{BE} : \mathrm{EC} = 12 : 6$
EF ∥ CD gives $\triangle \mathrm{BEF} \sim \triangle \mathrm{BCD}$ with ratio $\mathrm{BE} : \mathrm{BC}$ :
$\mathrm{EF} = 6 \times \frac{12}{18} = 4$

Q1BasicGr 9 · Term 3Similar figures

$\triangle \mathrm{ABC} \sim \triangle \mathrm{DEF}$ with $\mathrm{AB} = 8\text{ cm}$ and $\mathrm{DE} = 20\text{ cm}$ . Give each ratio in lowest terms.

(1)the ratio of similarity of $\triangle \mathrm{ABC}$ to $\triangle \mathrm{DEF}$
(2)the ratio of the areas of $\triangle \mathrm{ABC}$ and $\triangle \mathrm{DEF}$

Q2StandardGr 9 · Term 3Similar figures

In the figure, lines $\ell,\ m,\ n$ are parallel. Find $x$ .

Q3AdvancedGr 9 · Term 3Similar figures

In $\triangle \mathrm{ABC}$ , $\mathrm{DE} \parallel \mathrm{BC}$ and $\mathrm{AD} : \mathrm{DB} = 1 : 2$ . The area of $\triangle \mathrm{ADE}$ is 4 cm $^2$ . Find the area of quadrilateral DBCE.

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Similar figures: practice problems

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Related topics

Triangles and quadrilaterals

The Pythagorean theorem