Question

△abc and △def are similar.
(a) there are three proportions below. the ratios in each proportion compare the length of
one of the sides of △abc to the length of the corresponding side of △def.
complete the proportions.
\\( \frac { a b } { \square } = \frac { 2 } { 3 } \\)
\\( \frac { a c } { d f } = \square \\)
\\( \frac { b c } { \square } = \square \\)
(b) choose the correct statement about the answers to part (a).
oeach 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.
oeach 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.
oeach pair of side lengths compared is not in the same ratio. this is because in a
proportion both ratios must be different.
oeach pair of side lengths compared is not in the same ratio. this is because the
triangles are not right triangles.

Explanation:

Step1: Find the denominator for \(\frac{AB}{\square}\)

Since \(\triangle ABC\) and \(\triangle DEF\) are similar, and \(AB = 6\), \(DE=9\). The ratio \(\frac{AB}{DE}=\frac{6}{9}=\frac{2}{3}\).

Step2: Calculate \(\frac{AC}{DF}\)

Given \(AC = 8\), \(DF = 12\). Then \(\frac{AC}{DF}=\frac{8}{12}=\frac{2}{3}\).

Step3: Calculate \(\frac{BC}{\square}\)

Given \(BC = 10\), \(EF = 15\). Then \(\frac{BC}{EF}=\frac{10}{15}=\frac{2}{3}\).

For part (b), since the triangles are similar, corresponding sides are in proportion.

Answer:

a) \(\frac{AB}{DE}=\frac{2}{3}\), \(\frac{AC}{DF}=\frac{2}{3}\), \(\frac{BC}{EF}=\frac{2}{3}\)

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.