Chapter 9, Problem 2STP

To determine

Find the length of the base of the triangle.

Answer to Problem 2STP

Base of the triangle is 20 cm. Option J is correct.

Explanation of Solution

Given:

The area of a triangle in which the length of the base is 4 cm greater than twice the height is 80 square centimeters. What is the length of the base of the triangle?

Concept Used:

Translate the conditions given in the question into equations.

Let the height be is x cm

Length of the base is 4 cm greater than twice the height = 2x + 4

Area = 80 square centimeter.

Area of the Triangle is : $\text{A}=\frac{1}{2}·\text{base}·\text{height}$

The equation: $\text{80}=\frac{1}{2}·\text{(2x+4)}·\text{x}$

Calculation:

Let the height be is x cm

Length of the base is 4 cm greater than twice the height = 2x + 4

Area = 80 square centimeter.

Area of the Triangle is : $\text{A}=\frac{1}{2}·\text{base}·\text{height}$

The equation: $\text{80}=\frac{1}{2}·\text{(2x+4)}·\text{x}$

Solve for x:

  $\begin{array}{l}\text{A}=\frac{1}{2}·\text{base}·\text{height}\\ \text{80}=\frac{1}{2}·\text{(2x+4)}·\text{x}\\ \text{2}·\text{80}\text{=}\text{2}·\frac{1}{2}·\text{(2x+4)}·\text{x}\text{[}\text{multiply}\text{each}\text{side}\text{by}\text{2}\text{]}\\ \text{160}\text{=}{\text{2x}}^2+4x\\ \frac{160}{2}=\frac{{\text{2x}}^2+4x}{2}[\text{divide}\text{each}\text{side}\text{by}\text{2}\text{]}\\ \text{80}\text{=}{\text{x}}^2+2x\\ {\text{x}}^2+2x-80=0\text{[}\text{subtract}\text{80}\text{from}\text{both}\text{sides}]\\ (x+10)(x-8)=0[\text{factor}\text{the}\text{Trinominals}]\\ \text{Use}\text{zero}\text{product}\text{rule}\text{:}\\ \text{x+10}\text{=}\text{0}\text{x}\text{-}\text{8}\text{=}\text{0}\\ \text{x}\text{=}\text{-}\text{10}\text{x}\text{=}\text{8 }\end{array}$

Height cannot be negative, so x = 8

Height = 8 cm , base = 2x + 4 = 2 (8) + 4 = 20 cm

Thus, Base of the triangle is 20 cm. Option J is correct.