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A square is inscribed in a right triangle, as shown below. Find the area of the square.
 

 

I don't know how to add an image but the right triangle as a base of 1 and a height of 3.

 Jun 10, 2024
edited by ABJelly  Jun 10, 2024
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Line EF is parallel to line BC, due to both being lines of the square. Therefore, angle FEA is equal to angle ACB and angle AFE is equal to angle ABC. We can now prove that triangle AFE is similar to triangle ABC due to AA similarity. Let's suppose that the side length of the square is x. Using the ratios of the similar triangles:

\(\frac{3-x}{x} = \frac{3}{1}\)

\(3-x = 3x\)

\(4x=3\)

\(x=\frac{3}{4}\)

So the side length of the square is 3/4. Therefore, the area of the square is (3/4)^2 = 9/16 square units.

 Jun 10, 2024

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