Question

The base of a triangle is seven less than twice its height. If the area of the triangle is 102cm squared, find the length of the base.

Answer 1

length of base = 17 cm

Explanation:

Area of a triangle A = 1/2 bh where b = base h = height

We are given
b = 2h - 7

A = 102

or
2h = b + 7

h = (b+7) /2

use this in the equation for A

1/2 (b) (b+7)/2 = 102

1/4 x b(b+7) = 102

b(b+7) = 4 x 102

b² + 7 b = 408

b² + 7b - 408 = 0

This is a quadratic equation which can be solved by factoring or using the quadratic formula

We can factor b² + 7b - 408 as follows

• Find factors u, v of -408 (one factor positive, the other negative)
• See which of these factors will add(or subtract) to 7

• Factors of -408 are
  24 x -17 => adding we get 24 - 17 = 7
  

• So factoring b² + 7b - 408 = 0 we get
  (b-17)(b + 24) = 0
  
• This means b - 17 = 0 or b + 24 = 0

• b - 17 = 0 => b = 17

• b + 24 = 0 => b = -24

• Disregard negative value to get
  b = 17 cm

Check if b = 17 is correct

With b = 17

h = (b+7) /2

= (17 + 7)/2

= 24/2

= 12

Therefore A = 1/2 x 17 x 12

= 17 x 6

= 102