MathLabs

Problem 4

Given a plane π\pi, a point PP in π\pi and a point QQ not in π\pi, find all points RR in π\pi such that the ratio QP+PRQR\dfrac{QP+PR}{QR} is a maximum.
Step 4 of 5: Maximize by a right angle
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QP+PRQR=SRQR=sin⁡∠RQSsin⁡∠QSR\dfrac{QP+PR}{QR}=\dfrac{SR}{QR}=\dfrac{\sin\angle RQS}{\sin\angle QSR}
Detailed analysis

The angle QSR is fixed because S and R lie on a fixed ray from S. By the sine rule, the ratio is maximized when sin(angle RQS) is 1, namely when angle RQS=90 degrees.