Find length and width of a triangle

The Phillips family wants to fence their backyard. They know the yard has a perimeter of 24 meters and an area of 32 square meters. What is the yard's length and width?

Assuming that the yard is a rectangle

Area = length × width
32 = length × width

Perimeter = 2 × length + 2 × width

24 = 2 × length + 2 × width

Using 32 = length × width, width = 32 / length

Replace width = 32 / length in 24 = 2 × length + 2 × width


We get:

24 = 2 × length + 2 × (32 / length)

24 = 2length + 64 / length

24 - 2length = 64 / length

Multiply both sides by length

24 × length - 2length × length = ( 64 × length ) / length


24length - 2length² = 64


24length - 24length - 2length² = 64 - 24length

0 - 2length² = + 64 - 24length

0 - 2length² + 2length² = 64 - 24length + 2length²


0 + 0 + = 64 - 24length + 2length²

2length² - 24length + 64 = 0

Divide both sides by 2

length² - 12length + 32 = 0

Look for factors of 32 that will add up to -12

-8 × -4 = 32 and -8 + -4 = -12

So our factors are -8 and -4

We get (length + -8) × (length + -4) = 0

length + -8 = 0

length + -4 = 0


length = 8

length = 4


If length = 8, width = 32 / length = 32 / 8 = 4

If length = 4, width = 32 / length = 32 / 4 = 8


Thus, either the length = 4 and the width = 8 or the length = 8 and the width = 4