JEE Advanced Maths: Triangles (Similarity) questions with solutions

Q1. In a figure, angle QPS = angle RPT and angle PST = angle PQR. Prove that triangle PST is similar to triangle PQR, and hence find the ratio ST: QR given PR: QR = 4: 5. (Find ST: QR.)

1. ST: QR = 4: 5
2. ST: QR = 5: 4
3. ST: QR = 16: 25
4. ST: QR = 2: 5

Answer: ST: QR = 4: 5

Since angle QPS = angle RPT, adding angle SPR to both sides gives angle QPR = angle SPT. Also angle PST = angle PQR (given). By AA similarity, triangle PST ~ triangle PQR. Corresponding sides: ST/QR = PT/PR = PS/PQ. From the correspondence P<->P, S<->Q, T<->R, ST corresponds to QR and PT corresponds to PR. Given the ratio of corresponding sides equals PR-related ratio 4:5, ST: QR = 4: 5.