Question

similarity
relationships about ratios within and between
(a) there are three proportions below. the ratios in each proportion compare the length of
one of the sides of δabc to the length of the corresponding side of δpqr. complete
the proportions.
\\(\frac{ab}{pq}=\square\\) \\(\frac{ac}{?}=\square\\) \\(\frac{bc}{?}=\frac{2}{3}\\)
(b) choose the correct statement about the answers to part (a).
\\(\circ\\)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.
\\(\circ\\)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.
\\(\circ\\)each pair of side lengths compared is not in the same ratio. this is because the
triangles are not right triangles.
\\(\circ\\)each pair of side lengths compared is not in the same ratio. this is because in a
proportion both ratios must be different.

Explanation:

Part (a)

First Proportion: $\boldsymbol{\frac{AB}{PQ}}$

Step1: Identify side lengths

$AB = 8$, $PQ = 12$.

Step2: Calculate the ratio

$\frac{AB}{PQ} = \frac{8}{12} = \frac{2}{3}$.

Second Proportion: $\boldsymbol{\frac{AC}{?}}$

Step1: Identify corresponding sides

$AC = 10$, the corresponding side in $\triangle PQR$ is $PR = 15$.

Step2: Calculate the ratio

$\frac{AC}{PR} = \frac{10}{15} = \frac{2}{3}$.

Third Proportion: $\boldsymbol{\frac{BC}{?} = \frac{2}{3}}$

Step1: Identify corresponding sides

$BC = 12$, the corresponding side in $\triangle PQR$ is $QR = 18$.

Step2: Verify the ratio

$\frac{BC}{QR} = \frac{12}{18} = \frac{2}{3}$.

Part (b)

For similar triangles, corresponding sides are in proportion (same ratio). The ratios from part (a) are all $\frac{2}{3}$, which is a property of similar triangles (corresponding sides of similar triangles are proportional). The first option is wrong as it's not a coincidence. The third and fourth options are wrong as the ratios are the same. The second option correctly states the reason (similar triangles have proportional corresponding sides).

Answer:

• (a) $\frac{AB}{PQ} = \boldsymbol{\frac{2}{3}}$; $\frac{AC}{\boldsymbol{PR}} = \boldsymbol{\frac{2}{3}}$; $\frac{BC}{\boldsymbol{QR}} = \frac{2}{3}$
• (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.