A line intersects sides PQ and PR of ∆ PQR at A and B, respectively, and is parallel to QR, as shown in the figure. Prove that AP/PQ= BR/PR
Question
A line intersects sides PQ and PR of ∆ PQR at A and B, respectively, and is parallel to QR, as shown in the figure. Prove that
Solution
To prove AP/PQ = BR/PR, we can use the concept of similar triangles.
Step 1: Identify the triangles In the given figure, we can identify two triangles: Triangle APQ and Triangle BRP.
Step 2: Prove the triangles are similar Since the line AB is parallel to QR, by the Alternate Interior Angles Theorem, we know that ∠PAQ = ∠BRP and ∠APQ = ∠BPR. Therefore, by the Angle-Angle (AA) criterion, the two triangles are similar.
Step 3: Use the properties of similar triangles In similar triangles, the ratio of the lengths of corresponding sides is equal. Therefore, we have AP/PQ = BR/PR.
So, we have proved that AP/PQ = BR/PR.
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