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 $\triangle def$. complete the proportions.
$\frac{ab}{\text{?}} = \frac{1}{4}$ $\frac{ac}{df} = \square$ $\frac{bc}{\text{?}} = \square$

(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 the triangles are not right triangles.
\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 in the same ratio. this is coincidence. we would usually not expect this from similar triangles that are not the same size.
\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

Explanation:

Part (a)

First Proportion: $\boldsymbol{\frac{AB}{?} = \frac{1}{4}}$

• From the triangles, $AB = 3$ and $DE = 12$. Let's check the ratio: $\frac{AB}{DE} = \frac{3}{12} = \frac{1}{4}$. So the missing side is $DE$.

Second Proportion: $\boldsymbol{\frac{AC}{DF} = \square}$

• $AC = 5$ and $DF = 20$. Calculate the ratio: $\frac{AC}{DF} = \frac{5}{20} = \frac{1}{4}$.

Third Proportion: $\boldsymbol{\frac{BC}{?} = \square}$

• $BC = 6$ and $EF = 24$. Calculate the ratio: $\frac{BC}{EF} = \frac{6}{24} = \frac{1}{4}$. So the missing side is $EF$ and the ratio is $\frac{1}{4}$.

Part (b)

• Similar triangles have corresponding sides in proportion (same ratio). The triangles $\triangle ABC$ and $\triangle DEF$ have corresponding sides with ratio $\frac{1}{4}$, so they are similar. The correct statement is the one explaining that corresponding sides of similar triangles are in the same ratio.

Final Answers

(a)

• $\frac{AB}{DE} = \frac{1}{4}$
• $\frac{AC}{DF} = \frac{1}{4}$
• $\frac{BC}{EF} = \frac{1}{4}$

(b)

The correct option is: "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."

Answer:

Part (a)

First Proportion: $\boldsymbol{\frac{AB}{?} = \frac{1}{4}}$

• From the triangles, $AB = 3$ and $DE = 12$. Let's check the ratio: $\frac{AB}{DE} = \frac{3}{12} = \frac{1}{4}$. So the missing side is $DE$.

Second Proportion: $\boldsymbol{\frac{AC}{DF} = \square}$

• $AC = 5$ and $DF = 20$. Calculate the ratio: $\frac{AC}{DF} = \frac{5}{20} = \frac{1}{4}$.

Third Proportion: $\boldsymbol{\frac{BC}{?} = \square}$

• $BC = 6$ and $EF = 24$. Calculate the ratio: $\frac{BC}{EF} = \frac{6}{24} = \frac{1}{4}$. So the missing side is $EF$ and the ratio is $\frac{1}{4}$.

Part (b)

• Similar triangles have corresponding sides in proportion (same ratio). The triangles $\triangle ABC$ and $\triangle DEF$ have corresponding sides with ratio $\frac{1}{4}$, so they are similar. The correct statement is the one explaining that corresponding sides of similar triangles are in the same ratio.

Final Answers

(a)

• $\frac{AB}{DE} = \frac{1}{4}$
• $\frac{AC}{DF} = \frac{1}{4}$
• $\frac{BC}{EF} = \frac{1}{4}$

(b)

The correct option is: "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."