Question

(a) there are three proportions below. the ratios in each proportion compare the length of one of the sides of $\triangle abc$ to the length of the corresponding side of $\triangle xyz$. complete the proportions.
$\frac{ab}{\boldsymbol{?}} = \boldsymbol{\square}$
$\frac{ac}{\boldsymbol{?}} = \frac{1}{2}$
$\frac{bc}{yz} = \boldsymbol{\square}$
(b) choose the correct statement about the answers to part (a).
\bigcirc each pair of side lengths compared is not in the same ratio. this is because in a proportion both ratios must be different.
\bigcirc each pair of side lengths compared is not in the same ratio. this is because the triangles are not right triangles.
\bigcirc each pair of side lengths compared is in the same ratio. this is because the side lengths in each pair are lengths of corresponding sides and the triangles are similar.
\bigcirc each pair of side lengths compared is in the same ratio. this is coincidence. we would usually not expect this from similar triangles that are not the same size.

Explanation:

Part (a)

First Proportion ($\boldsymbol{\frac{AB}{?} = \square}$)

Step1: Identify Corresponding Sides

In $\triangle ABC$, $AB = 5$. In $\triangle XYZ$, the corresponding side to $AB$ is $XY = 10$.

Step2: Calculate the Ratio

The ratio is $\frac{AB}{XY}=\frac{5}{10}=\frac{1}{2}$. So the denominator is $XY$ and the ratio is $\frac{1}{2}$.

Second Proportion ($\boldsymbol{\frac{AC}{?} = \frac{1}{2}}$)

Step1: Identify Corresponding Sides

In $\triangle ABC$, $AC = 8$. Let the corresponding side in $\triangle XYZ$ be $XZ = 16$.

Step2: Verify the Ratio

$\frac{AC}{XZ}=\frac{8}{16}=\frac{1}{2}$. So the denominator is $XZ$.

Third Proportion ($\boldsymbol{\frac{BC}{YZ} = \square}$)

Step1: Identify Side Lengths

$BC = 11$ and $YZ = 22$.

Step2: Calculate the Ratio

$\frac{BC}{YZ}=\frac{11}{22}=\frac{1}{2}$.

Part (b)

To determine the correct statement, we analyze the ratios from part (a). All the ratios of corresponding sides ( $\frac{AB}{XY}=\frac{1}{2}$, $\frac{AC}{XZ}=\frac{1}{2}$, $\frac{BC}{YZ}=\frac{1}{2}$ ) are equal. By the definition of similar triangles, if the ratios of corresponding sides of two triangles are equal, the triangles are similar. So each pair of side lengths (corresponding sides) is in the same ratio because the triangles are similar.

Answer:

Each pair of side lengths compared is in the same ratio. This is because the side lengths in each pair are lengths of corresponding sides and the triangles are similar.

Final Answers for Part (a)

• $\frac{AB}{XY}=\boldsymbol{\frac{1}{2}}$
• $\frac{AC}{XZ}=\boldsymbol{\frac{1}{2}}$
• $\frac{BC}{YZ}=\boldsymbol{\frac{1}{2}}$

Final Answer for Part (b)

Each pair of side lengths compared is in the same ratio. This is because the side lengths in each pair are lengths of corresponding sides and the triangles are similar.