Question

In triangles ABC and PQR, \( \angle A = \angle Q \) and \( \angle B = \angle R \), then \( AB : AC \) is equal to :

• A. \( PQ : PR \)
• B. \( PQ : QR \)
• C. \( QR : QP \)
• D. \( PR : QR \)

Hint:

The order of vertices in a similarity statement is crucial. Match the equal angles to set up the correct ratio.

Answer 1

The Correct Option is C

To solve the problem, we need to analyze the information given about the triangles \( \triangle ABC \) and \( \triangle PQR \). We are told that \( \angle A = \angle Q \) and \( \angle B = \angle R \). This suggests that triangles ABC and PQR are similar based on the Angle-Angle (AA) similarity criterion. When two triangles are similar by the AA criterion, their corresponding sides are proportional.

Thus, the corresponding sides are \( AB \) with \( QR \), \( AC \) with \( QP \), and \( BC \) with \( PR \). We need to find the ratio \( AB : AC \) and compare it with the options given.

The ratios of corresponding sides for similar triangles are equal. Therefore, we have:

\(\frac{AB}{QR} = \frac{AC}{QP} = \frac{BC}{PR}\)

From the question, we are interested in \( AB : AC \). Thus, we compare it with the corresponding ratio \( QR : QP \):

\(\frac{AB}{AC} = \frac{QR}{QP}\)

Hence, \( AB : AC \) is equal to \( QR : QP \).

Therefore, the correct option is: \( QR : QP \).