Question

△abc and △pqr are similar. find the missing side length.

Explanation:

Step1: Identify corresponding sides

In similar triangles, corresponding sides are proportional. Let's assume \( \triangle ABC \sim \triangle PQR \), so \( \frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR} \). We know \( AC = 7 \), \( PR = 35 \), \( BC = 8 \), \( QR = 40 \), and \( AB = 4 \), we need to find \( PQ \). First, find the scale factor using \( AC \) and \( PR \). The scale factor \( k=\frac{PR}{AC}=\frac{35}{7} = 5 \).

Step2: Use scale factor to find PQ

Since the scale factor is 5, and \( AB \) corresponds to \( PQ \), we can also use the ratio of \( BC \) and \( QR \) to confirm the scale factor ( \( \frac{40}{8}=5 \) ). Then, \( PQ = AB\times k \)? Wait, no, wait. Wait, actually, if \( \triangle ABC \sim \triangle PQR \), then the ratio of \( AB \) to \( PQ \) should be equal to the ratio of \( AC \) to \( PR \). So \( \frac{AB}{PQ}=\frac{AC}{PR} \). Plugging in the values: \( \frac{4}{PQ}=\frac{7}{35} \). Cross - multiply: \( 7\times PQ=4\times35 \). Then \( 7PQ = 140 \), divide both sides by 7: \( PQ=\frac{140}{7}=20 \). Wait, or maybe I mixed up the correspondence. Wait, let's check the sides. \( AC = 7 \), \( PR = 35 \); \( BC = 8 \), \( QR = 40 \). So \( \frac{AC}{PR}=\frac{7}{35}=\frac{1}{5} \), \( \frac{BC}{QR}=\frac{8}{40}=\frac{1}{5} \). So the scale factor from \( \triangle ABC \) to \( \triangle PQR \) is 5. So \( AB = 4 \), so \( PQ=AB\times5 = 20 \). Alternatively, using proportion: \( \frac{AB}{PQ}=\frac{AC}{PR} \), so \( \frac{4}{x}=\frac{7}{35} \), solve for \( x \). Cross multiply: \( 7x = 4\times35 \), \( 7x = 140 \), \( x = 20 \).

Answer:

20