Similar triangles

Geometry Level 3

In the diagram below, the top side of a square is extended to the right, the end of which is then connected to the lower left vertex of the square. This line segment is divided into two parts: the part with length 30 inside the square and the part with length 10 outside.

What is the side length of the square?

The answer is 24.

1 solution

A Former Brilliant Member
Jul 10, 2017

Relevant wiki: Similar Triangles Problem Solving - Basic

Let $x=AB$ .

Since $\triangle CEF \sim \triangle DEA$ , we have, $\dfrac{CE}{ED}=\dfrac{10}{30}=\dfrac{1}{3}$ $\implies$ $ED=3(CE)$

That means $ED=\dfrac{3}{4}x$ . Applying Pythagorean theorem on $\triangle DEA$ , we have

$x^2+\left(\dfrac{3}{4}x\right)^2=30^2$

$x^2+\dfrac{9}{16}x^2=900$

$\dfrac{25}{16}x^2=900$

$x^2=\dfrac{16}{25}(900)$

$x^2=576$

$x=\sqrt{576}=$ $\large \color{#D61F06}\boxed{24}$

i thought about this solution too, but i ended up solving this system : (h + x)^2 +h^2 = 1600, y^2 + x^2 = 100, h^2 + (h-y)^2 = 900, where h is the side lenght of the square x,y the sides of hypotenuse 10 triangle.

aaaaaaa aaaaaa - 3 years, 11 months ago

Your solution is also correct, but it is longer.

A Former Brilliant Member - 3 years, 11 months ago

Similar triangles

The answer is 24.

1 solution

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