Maths Olympiad Prep

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Problem 1089

AMC 12 late, AIME early
Geometry Difficulty 4.7 Find the answer HMMT February

Let (x,y)(x, y) be a point in the cartesian plane, x,y>0x, y>0. Find a formula in terms of xx and yy for the minimal area of a right triangle with hypotenuse passing through (x,y)(x, y) and legs contained in the xx and yy axes.

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

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Official solution

2xy2 x y.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.