Question

The base of a triangle exceeds the height by 9 feet. If the area is 180 square feet, find the length of the base and the height of the triangle.

Answer 1

Area of a triangle = (1/2) *base*height

height = x
base = x+9

Area =(1/2)*(x+9)*x
180= (1/2)*(x+9)*x
360= (x+9)*x
x²+9x-360 =0

a=1, b=9, c=-360
D=b²-4ac=81+4*360=1521, √D=√1521=39

x=(-b+/-√D)/2a
x=(-9+/-39)/2
We need only positive solution,
x=(-9+39)/2
x=15 height
x+9=15+9=24 base

Height -15 ft, base - 24 feet.

Answer 2

base = x + 9
height = x
Area = 1/2 · b · h
180 = 1/2 (x + 9)(x)
180 = (x² + 9x)/2
360 = x² + 9x
0 = x² + 9x - 360

Use Quadratic Formula to solve:
x = (-b +/- √b² - 4ac)/(2a)
= (-9 +/- √9² - 4·1·-360)/(2·1)
= (-9 +/- √9² + 1440)/(2)
= (-9 +/- √1449)/(2)
= (-9 +/- 38)/(2)
= (-9 + 38)/(2) , (-9 - 38)/(2)
= 29/2 , -47/2
= 14.5 , -23.5
length cannot be negative!!! so disregard the -23.5

Answer: The height of the triangle is 14.5 ft and the base is 14.5 + 9 = 23.5 ft