overlapping shapes, a triangle and a square

Question

the ratio of the area of the triangle to the area of the square is 5:2. the ratio of the unshaded part of the triangle to the unshaded part of the square is 5:1. find the area of the shaded part if the length of each side of the squarwe is 4 inches?

Answer 1

 

Hi Liz,

 

If we denote each area as follows,

 

     - the unshaded part of the triangle:   x

     - the unshaded part of the square:    y

     - the shaded part of the triangle

        = the shaded part of the square:    z,

 

the problem is to find out the value of z

when

 

     (x+z) : (y+z) = 5 : 2            (Eq. 1)

     x : y = 5 : 1                         (Eq. 2)

     y + z = 16                           (Eq. 3)

 

From Eq. 1,

 

     2(x+z) = 5(y+z)

     2x = 5y + 3z

     x = (5/2)y + (3/2)z              (Eq. 4)

 

From Eq. 2,

 

     x = 5y                                 (Eq. 5)

 

From Eq. 4 and Eq. 5,

 

     5y = (5/2)y + (3/2)z

     (5/2)y - (3/2)z = 0

     5y - 3z = 0                          (Eq. 6)

 

Solving a system of Eq. 3 and Eq. 6, we get z = 10.

 

The final answer is 10 square-inches.