Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Find the answer

3. Given points $A(1,2)$ and $B(3,4)$ , there is a point $P$ on the coordinate axis, and $P A + P B$ is minimized. Then the coordinates of point $P$ are $\qquad$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

3. $\left(0, \frac{5}{2}\right)$ .

Using knowledge of axis symmetry, ruler and compass constructions, and linear functions, we know that the minimum value of $P A + P l 3$ occurs when $P$ is on the $y$ -axis, which is smaller than when $P$ is on the $x$ -axis. Thus, the coordinates of $P$ are $\left(0, \frac{5}{2}\right)$ .

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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.