Question

(a) complete the three pairs of proportions below.
ratio of lengths of sides of δabc
ratio of the lengths of sides of δxyz that correspond to the sides of δabc in the ratio above
\\(\frac{ab}{ac}=\square\\)
\\(\frac{?}{xz}=\frac{4}{3}\\)
\\(\frac{ab}{bc}=\square\\)
\\(\frac{?}{yz}=2\\)
\\(\frac{ac}{bc}=\frac{3}{2}\\)
\\(\frac{xz}{?}=\square\\)
(b) choose the correct statement about the answers to part (a).
\\(\circ\\)in each pair of proportions, the lengths of the sides of δabc are in the same ratio as the lengths of the sides of δxyz. this is because we compared the sides of δxyz that correspond to the sides of δabc and the triangles are similar.
\\(\circ\\)in each pair of proportions, the lengths of the sides of δabc are in the same ratio as the lengths of the sides of δxyz. this is coincidence. we would usually not expect this from similar triangles that are not the same size

Explanation:

Part (a)

First Proportion: $\boldsymbol{\frac{AB}{AC}}$ and Corresponding $\boldsymbol{\triangle XYZ}$ Side

Step1: Calculate $\frac{AB}{AC}$

In $\triangle ABC$, $AB = 12$ and $AC = 9$. So, $\frac{AB}{AC}=\frac{12}{9}=\frac{4}{3}$.
The corresponding side in $\triangle XYZ$ for $AB$ is $XY$ (since the triangles are similar, corresponding sides are proportional). So the ratio is $\frac{XY}{XZ}=\frac{4}{3}$.

Step2: Fill in the blanks

For $\frac{AB}{AC}$, we have $\frac{12}{9}=\frac{4}{3}$. The corresponding ratio for $\triangle XYZ$ is $\frac{XY}{XZ}=\frac{4}{3}$.

Second Proportion: $\boldsymbol{\frac{AB}{BC}}$ and Corresponding $\boldsymbol{\triangle XYZ}$ Side

Step1: Calculate $\frac{AB}{BC}$

In $\triangle ABC$, $AB = 12$ and $BC = 6$. So, $\frac{AB}{BC}=\frac{12}{6}=2$.
The corresponding side in $\triangle XYZ$ for $AB$ is $XY$ and for $BC$ is $YZ$. So the ratio is $\frac{XY}{YZ}=2$.

Step2: Fill in the blanks

For $\frac{AB}{BC}$, we have $\frac{12}{6}=2$. The corresponding ratio for $\triangle XYZ$ is $\frac{XY}{YZ}=2$.

Third Proportion: $\boldsymbol{\frac{AC}{BC}}$ and Corresponding $\boldsymbol{\triangle XYZ}$ Side

Step1: Calculate $\frac{XZ}{YZ}$

We know $\frac{AC}{BC}=\frac{3}{2}$. In $\triangle XYZ$, $XZ = 15$ and $YZ = 10$. So, $\frac{XZ}{YZ}=\frac{15}{10}=\frac{3}{2}$.

Step2: Fill in the blanks

For $\frac{AC}{BC}=\frac{3}{2}$, the corresponding ratio for $\triangle XYZ$ is $\frac{XZ}{YZ}=\frac{15}{10}=\frac{3}{2}$.

Part (b)

Similar triangles have corresponding sides in proportion. This is a property of similar triangles, not a coincidence. The first statement correctly explains that the ratios are equal because the triangles are similar and we compared corresponding sides. The second statement is incorrect as the proportionality is due to similarity, not coincidence.

Answer:

In each pair of proportions, the lengths of the sides of $\triangle ABC$ are in the same ratio as the lengths of the sides of $\triangle XYZ$. This is because we compared the sides of $\triangle XYZ$ that correspond to the sides of $\triangle ABC$ and the triangles are similar.

Final Answers for Part (a)

• $\frac{AB}{AC}=\boldsymbol{\frac{4}{3}}$, and $\frac{\boldsymbol{XY}}{XZ}=\frac{4}{3}$
• $\frac{AB}{BC}=\boldsymbol{2}$, and $\frac{\boldsymbol{XY}}{YZ}=2$
• $\frac{XZ}{\boldsymbol{YZ}}=\boldsymbol{\frac{3}{2}}$ (since $\frac{AC}{BC}=\frac{3}{2}$ and $\frac{XZ}{YZ}=\frac{15}{10}=\frac{3}{2}$)

Final Answer for Part (b)

The correct statement is: "In each pair of proportions, the lengths of the sides of $\triangle ABC$ are in the same ratio as the lengths of the sides of $\triangle XYZ$. This is because we compared the sides of $\triangle XYZ$ that correspond to the sides of $\triangle ABC$ and the triangles are similar."