Question

In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R, then AB : AC is equal to ______.

पर्याय

• PQ : PR

• PQ : QR

• QR : QP

• PR : QR

Answer 1

In triangles ABC and PQR, ∠A = ∠Q and ∠B = ∠R, then AB : AC is equal to QR : QP.

Explanation:

1. Identify Similarity

In triangles ΔABC and ΔQRP, we are given

∠A = ∠Q

∠B = ∠R

Since two angles are equal, the triangles are similar by the AA (Angle-Angle) similarity criterion:

ΔABC ∼ ΔQRP

2. Determine Proportional Sides

When two triangles are similar, the ratios of their corresponding sides are equal. Based on the vertex correspondence (A → Q, B → R, C → P), we have:

`(AB)/(QR) = (BC)/(RP) = (AC)/(QP)`

3. Find the Ratio

To find the value of AB : AC, we use the first and last parts of the proportion:

`(AB)/(QR) = (AC)/(QP)`

Rearranging the terms to isolate `(AB)/(AC)`:

`(AB)/(AC) = (QR)/(QP)`

Therefore, AB : AC = QR : QP.