Question

determine if triangle pqr and triangle stu are or are not similar, and, if they are, state how you know. (note that figures are not necessarily drawn to scale.)

triangle pqr has sides: pr = 15, pq = 11, qr = 17.
triangle stu has sides: su = 75, st = 55, tu = 85.

answer attempt 1 out of 3

the triangles dropdown similar.

Explanation:

Step1: Check Triangle PQR Sides

Triangle \( PQR \): \( PR = 15 \), \( PQ = 11 \), \( QR = 17 \). Wait, no—wait, \( PQ = 11 \)? Wait, no, maybe I misread. Wait, no, let's check triangle \( STU \): \( SU = 75 \), \( ST = 55 \), \( UT = 85 \). Wait, triangle \( PQR \): sides are 15, 11, 17? Wait, no, maybe \( PQ \) is 8? Wait, no, 15-8-17 is a Pythagorean triple (since \( 15^2 + 8^2 = 225 + 64 = 289 = 17^2 \)). Wait, the given \( PQ \) is 11? That can't be. Wait, maybe a typo? Wait, no, the user's image: triangle \( PQR \) has \( PR = 15 \), \( QR = 17 \), \( PQ = 8 \)? Wait, no, the user wrote \( PQ = 11 \)? Wait, no, maybe I misread. Wait, let's check the ratios. Let's list the sides of both triangles in order (smallest to largest).

Triangle \( PQR \): Let's sort the sides. If \( PR = 15 \), \( PQ = 11 \), \( QR = 17 \). Wait, 11, 15, 17.

Triangle \( STU \): \( ST = 55 \), \( SU = 75 \), \( UT = 85 \). Sort: 55, 75, 85.

Now, check if the ratios are equal. Let's divide each side of \( STU \) by the corresponding side of \( PQR \) (if they are similar, the ratio should be constant).

First, 55 / 11 = 5.

75 / 15 = 5.

85 / 17 = 5.

Ah! So 55 ÷ 11 = 5, 75 ÷ 15 = 5, 85 ÷ 17 = 5. So all sides are in ratio 5:1. Therefore, by SSS (Side-Side-Side) similarity criterion, the triangles are similar because their corresponding sides are proportional.

Wait, earlier I thought \( PQ \) was 11, \( PR = 15 \), \( QR = 17 \). Then \( STU \) has \( ST = 55 \) (corresponding to \( PQ = 11 \)), \( SU = 75 \) (corresponding to \( PR = 15 \)), \( UT = 85 \) (corresponding to \( QR = 17 \)). So the ratios are 55/11 = 5, 75/15 = 5, 85/17 = 5. So all three sides are proportional, so SSS similarity.

Step2: Confirm SSS Similarity

SSS similarity states that if the corresponding sides of two triangles are in proportion, the triangles are similar. Here, the ratio of each pair of corresponding sides is 5 (55/11 = 5, 75/15 = 5, 85/17 = 5), so the sides are proportional. Thus, the triangles are similar by SSS similarity.

Answer:

The triangles are similar (by SSS similarity, as their corresponding sides are in the ratio 5:1).