Question

△abc and △pqr 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 △pqr. complete the proportions.
\\( \frac { a b } {? } = \frac { 3 } { 5 } \\) \\( \frac { a c } { p r } = \square \\) \\( \frac { b c } {? } = \square \\)
(b) choose the correct statement about the answers to part (a).
○ 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.
○ 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.
○ each pair of side lengths compared is not in the same ratio. this is because the triangles are not right triangles.
○ each pair of side lengths compared is not in the same ratio. this is because in a proportion both ratios must be different.

Explanation:

Step1: Find the corresponding side for \(AB\)

Since \(\triangle ABC\sim\triangle PQR\), the sides are in proportion. \(AB = 6\), and we know \(\frac{AB}{?}=\frac{3}{5}\). Let the corresponding side be \(x\). Then \(\frac{6}{x}=\frac{3}{5}\), cross - multiply gives \(3x = 30\), so \(x = 10\) (which is \(PQ\)).

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

Given \(AC = 15\), \(PR=25\). Then \(\frac{AC}{PR}=\frac{15}{25}=\frac{3}{5}\).

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

Given \(BC = 18\), \(QR = 30\). Then \(\frac{BC}{QR}=\frac{18}{30}=\frac{3}{5}\).

Answer:

(a) \(\frac{AB}{PQ}=\frac{3}{5}\), \(\frac{AC}{PR}=\frac{3}{5}\), \(\frac{BC}{QR}=\frac{3}{5}\)
(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.