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The diagram below consists of a small square, four equilateral triangles, and a large square.  Find the area of the large square.

 

 Feb 24, 2024
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The side length of each of the equilateral triangles is 1. 

We see that 2 of such side lengths almost make the side length of the larger square, but there's still a little gap.

Each gap is half of a equilateral triangle, or a 30-60-90 triangle, and the length we want is \(\frac{1}{2}\).

So the side length of the larger square is \(2+\frac{1}{2}=\frac{5}{2}\).

The area is \(({\frac{5}{2}})^{2}=\frac{25}{4}\). So the area of the larger square is 25/4.

 Feb 24, 2024

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