Question

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

Answer 1

5/9 of the area of square ABCD is shaded

Explanation:

see the attached figure to better understand the problem

we know that

To find out what fraction of the area of square ABCD is shaded, divide the shaded area by the total area of square ABCD

step 1

Find out the area of square ABCD

The area of a square is

`A=b^(2)`

where

b is the length side of the square

we have

`b=(x+2x)=3x\ units`

so

`A=(3x)^(2)`

`A=9x^2\ units^(2)`

step 2

Find out the area of the 4 congruent right triangles

`A=4[(1)/(2)(x)(2x)]=4x^(2)\ units^2`

step 3

Find out the area of the shaded region

The area of the shaded region is equal to the area of square ABCD minus the area of the 4 congruent right triangles

so

`A=9x^2-4x^(2)=5x^(2)\ units^(2)`

step 4

Divide the shaded area by the total area of square ABCD

`(5x^(2))/(9x^(2)) =(5)/(9)`

therefore

5/9 of the area of square ABCD is shaded