Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 2.3 Junior Find the answer

Given points $P(-1,-2)$ and $Q(4,2)$ in the $xy$ -plane; point $R(1,m)$ is taken so that $PR+RQ$ is a minimum. Then $m$ equals:

Pick one

Solution

By the Triangle Inequality, $PR + QR \ge PR$ , and equality holds if $R$ is on $PQ$ . The equation of the line with $P$ and $Q$ is $y = \frac{4}{5}x - \frac{6}{5}$ , so point $R$ is $(1,-\frac{2}{5})$ . Thus, $m = \boxed{\textbf{(B) } -\frac{2}{5}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.