Question

△abc and △xyz are similar.
(a) complete the three pairs of proportions below.
ratio of lengths of sides of △abc: $\frac{ab}{ac}=\frac{1}{2}$, $\frac{ab}{bc}=\frac{2}{5}$, $\frac{ac}{bc}=\frac{4}{5}$
ratio of the lengths of sides of △xyz that correspond to the sides of △abc in the ratio above: $\frac{xy}{xz}=\frac{1}{2}$, $\frac{?}{yz}=\frac{2}{5}$, $\frac{xz}{yz}=\frac{4}{5}$

Explanation:

Step1: Recall similarity property

For similar triangles, the ratios of corresponding - side lengths are equal.

Step2: Find the first proportion

We know that $\frac{AB}{AC}=\frac{6}{12}=\frac{1}{2}$. Since $\triangle ABC\sim\triangle XYZ$, the corresponding - side ratio $\frac{XY}{XZ}=\frac{10}{20}=\frac{1}{2}$.

Step3: Find the second proportion

We have $\frac{AB}{BC}=\frac{6}{15}=\frac{2}{5}$. The corresponding - side in $\triangle XYZ$ for $AB$ is $XY$ and for $BC$ is $YZ$. So, $\frac{XY}{YZ}=\frac{10}{25}=\frac{2}{5}$.

Step4: Find the third proportion

We have $\frac{AC}{BC}=\frac{12}{15}=\frac{4}{5}$. The corresponding - side in $\triangle XYZ$ for $AC$ is $XZ$ and for $BC$ is $YZ$. So, $\frac{XZ}{YZ}=\frac{20}{25}=\frac{4}{5}$.

Answer:

The completed proportions are: $\frac{XY}{XZ}=\frac{1}{2}$, $\frac{XY}{YZ}=\frac{2}{5}$, $\frac{XZ}{YZ}=\frac{4}{5}$