Question

A triangle with area of 28 square inches has a height that is two less than four times the base. Find the base and the height of the triangle. Base is _ inches Height is __ inches

Answer 1

We write the following equations from the data of the statement of the problem:

• the area of the triangle is A = 28,

,

• the height h and the base b are related by the following equation:

`h=4b-2`

The formula for the area of the triangle:

`A=(1)/(2)\cdot b\cdot h\text{.}`

Replacing the data of the problem in the equation above:

`28=(1)/(2)\cdot b\cdot(4b-2).`

We rewrite the equation in the following way:

`\begin{gathered} 2\cdot28=b\cdot(4b-2), \\ 56=4b^2-2b, \\ 4b^2-2b-56=0. \end{gathered}`

We have a quadratic equation for the length of base b, the solutions to this equation are:

`b=4\text{ and }b=-(7)/(2)\text{.}`

Because b is the length of one side of the triangle, and lengths are positive quantities, we must select the positive value of b, so we have:

`b=4.`

Replacing this result in the equation for the height, we get:

`h=4b-2=4\cdot4-2=16-2=14.`

Answer

• Base is ,4, inches,

,

• Height is, 14, inches.