Question

In the figure above points are chosen on each pair of adjacent sides of a square $ABCD$ to form $4$ congruent right triangles, as shown in the figure above. Each of these triangles has one leg that is twice as long as the other. Determine what fraction of the area of square $ABCD$ is shaded.

• A. $\frac{1}{9}$
• B. $\frac{2}{9}$
• C. $\frac{4}{9}$
• D. $\frac{5}{9}$
• E. $\frac{8}{9}$

Answer 1

The correct option is D $\frac{5}{9}$

Given, $ABCD$ is the outer square
and length of side of the square $ABCD$ is $2x+x=3x$
Area of bigger square is ${\left(side\right)}^2={\left(3x\right)}^2={9x}^2$
We can find the side of the shaded square with the help of pythagras theoram,the sides of shades square are forming hypotunes of a triangle.
We know,
Hypotunes $=\sqrt{{base}^2+{perpendicular}^2}$
$h=\sqrt{{\left(2x\right)}^2+{\left(x\right)}^2}=\sqrt{{4x}^2+x^2}=\sqrt{{5x}^2}$
So, side of shaded square is $=\sqrt{{5x}^2}$
Area of shaded square is ${\left(\sqrt{{5x}^2}\right)}^2={5x}^2$
So, fraction of the area of square $ABCD$ is shaded will be $\frac{{5x}^2}{{9x}^2}=\frac{5}{9}$