1Cademy - Finding the Third Sides of Similar Triangles Using Proportions

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Property of Similar Triangles

Example

Finding the Third Sides of Similar Triangles Using Proportions

Apply the geometry problem-solving strategy and the property of similar triangles to find the unknown third side of each triangle when two sides of each are known.

Problem: $\triangle ABC$ is similar to $\triangle XYZ$ . In the larger triangle, $AB = 4$ and $AC = 3.2$ . In the smaller triangle, $XY = 3$ and $YZ = 4.5$ . Find the lengths of side $BC = a$ and side $XZ = y$ .

Read the problem. The figure shows two similar triangles with the given side lengths labeled.
Identify: The lengths of the third side of each triangle.
Name: Let $a$ = the length of the third side of $\triangle ABC$ (side $BC$ ), and let $y$ = the length of the third side of $\triangle XYZ$ (side $XZ$ ).
Translate: Because the triangles are similar, their corresponding sides are proportional:

$\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}$

Since $AB = 4$ corresponds to $XY = 3$ , the known ratio is $\frac{AB}{XY} = \frac{4}{3}$ . Set up two proportions — one for each unknown — keeping the larger triangle's sides in the numerators and the smaller triangle's sides in the denominators:

To find $a$ : $\frac{4}{3} = \frac{a}{4.5}$

To find $y$ : $\frac{4}{3} = \frac{3.2}{y}$

Solve each proportion by cross-multiplying.

For $a$ : $3a = 4(4.5) = 18$ , so $a = 6$ .

For $y$ : $4y = 3(3.2) = 9.6$ , so $y = 2.4$ .

Check: For $a$ : $\frac{4}{3} \stackrel{?}{=} \frac{6}{4.5}$ → $4(4.5) = 18$ and $6(3) = 18$ ✓. For $y$ : $\frac{4}{3} \stackrel{?}{=} \frac{3.2}{2.4}$ → $4(2.4) = 9.6$ and $3.2(3) = 9.6$ ✓.
Answer: The third side of $\triangle ABC$ is $6$ and the third side of $\triangle XYZ$ is $2.4$ .

This example demonstrates a key strategy when working with similar triangles: first identify one pair of known corresponding sides to establish the ratio, then use that ratio to set up a separate proportion for each unknown side. Placing the sides of the larger triangle consistently in the numerators (or consistently in the denominators) helps avoid mismatched pairings.

Updated 2026-04-21

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OpenStax Elementary Algebra

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