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Harvard-MIT Mathematics Tournament

Geometry Difficulty 3.6 AMC 10/12 Find the answer United States

Problem:
A rectangle has perimeter $10$ and diagonal $\sqrt{15}$ . What is its area?

Your answer

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

Solution

Solution:
If the sides are $x$ and $y$ , we have $2x + 2y = 10$ , so $x + y = 5$ , and $\sqrt{x^{2} + y^{2}} = \sqrt{15}$ , so $x^{2} + y^{2} = 15$ .

Squaring the first equation gives $x^{2} + 2xy + y^{2} = 25$ , and subtracting the second equation gives $2xy = 10$ , so the area is $xy = 5$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.