Question

A portion of a rectangle is shaded as shown. The area of the shaded

region is 78 in.2. What is the value of x? Show your work.

Answer 1

Finding the area of the large rectangle.

`Area \ of \ Large \ rectangle = LB \\\\Area \ of \ Large \ rectangle = (12)(8)\\\\ Area \ of \ Large \ rectangle = 96 \ in^(2)`

We know that:

• 
  

This means that we got two ways to find the area of the triangle.

Those methods are:

• ⇒
  `Method \ 1 = (1)/(2) * Base * Height`
• ⇒
  `Method \ 2 = 96 - 78`

Let's put these two methods in an equation as both are equal.

• 
  `(1)/(2) * Base * Height = 96 - 78`

Substituting the base of the triangle:

• 
  `(1)/(2) * 12 * Height = 96 - 78`

Simplifying the RHS:

• 
  `(1)/(2) * 12 * Height = 18`

Simplifying the LHS:

• 
  `6 * Height = 18`

Dividing 6 both sides:

• 
  `Height = (18)/(6) = 3 \ in`

Since the height is 3 in, the measure of x is 5 in because the sum of the height and x must be 8.

Answer 2

x = 5 in

Explanation:

Area of a rectangle = width x height

⇒ area of rectangle = 12 x 8 = 96 in²

Given shaded area = 78 in²

Area of white triangle = area of rectangle - shaded area

= 96 - 78

= 18 in²

Area of a triangle = 1/2 x base x height

⇒ 18 = 1/2 x 12 x height

⇒ height = 3 in

If the height of the triangle is 3 in and the height of the rectangle is 8 in

⇒ 3 + x = 8

⇒ x = 5 in