Track / Stage 4 / 321 of 340 #581 of 1964

Problem 581

AMC 12 late, AIME early

Geometry Difficulty 5.0 Multiple choice

A rectangle with height $x$ and width $2 x$ has the same perimeter as an equilateral triangle with side length 10 . What is the area of the rectangle?
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Pick one

Official solution

Since the given equilateral triangle has side length 10 , its perimeter is $3 \times 10=30$ .

In terms of $x$ , the perimeter of the given rectangle is $x+2 x+x+2 x=6 x$ .

Since the two perimeters are equal, then $6 x=30$ which means that $x=5$ .

Since the rectangle is $x$ by $2 x$ , its area is $x(2 x)=2 x^{2}$ .

Since $x=5$ , its area is $2\left(5^{2}\right)=50$ .

ANSWER: (B)

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