Question

As shown in the figure, a square is drawn sharing one corner with a right triangle and the other three corners on the sides of this triangle. Calculate the length of a side of the square.

Answer 1

Let's name the corners as shown below.

Given : AD = 4 cm
FC = 3 cm
DEFB is a square

$\text{Let the length of each side of the}\text{square be}x.$

$\Rightarrow DE=EF=FB=DB=x$

$\text{Each interior angle of the square will}\text{be equal to}{90}^{\circ }\text{.}$

$\text{Since BDEF is a square,}DE||BF||BC\text{and}\angle ADE=\angle ABC$
$\text{(Corresponding angles are equal)}$

$\text{In}△ADE\text{and}△ABC,\angle ADE=\angle ABC\angle AED=\angle ACB$
$\text{(Corresponding angles are equal)}$

$\therefore △ADE\sim △ABC\text{(By AA similarity criterion)}$

$\Rightarrow$ $\frac{AD}{AB}=\frac{AE}{AC}=\frac{DE}{BC}$

$\text{Let's take}\frac{AD}{AB}=\frac{DE}{BC}$

$\Rightarrow$ $\frac{4}{4+x}=\frac{x}{x+3}$

$\Rightarrow$ $4x+12=4x+x^2$

$\Rightarrow$ $x^2=12$

$\therefore \text{Length of each side of the square,}x=\sqrt{12}=2\sqrt{3}cm$