Question

The base of a triangle is 10 inches more than 2 times the height. If the area of the triangle is 84 square inches, find the base and height?

Answer 1

`h = 7` ; `b = 24`

Explanation:

Substitute the value for b in h and then use the area formula to calculate each term.

`A = 1/2bxxh` ; `b = 2h+10`
`84 = 1/2(2h+10)xxh` ; `84 = h^2 + 5h`
`h = 7` ; `b = 24`

CHECK: `A = 1/2bxxh`; `84 = 1/2(24xx7)`

`84 = 1/2(168)` ; `84 = 84` Correct.

Answer 2

`"Height" = 7 "inches"`
`"Base" = 24 "inches"`

Explanation:

The formula for the area of a triangle is
`A = (1)/(2)(b*h)`

In this problem, you are not directly given the number of inches in the base, so find a way to express that value in terms of the height.

Find a way to express the number of inches in the base in terms of the height

Let `h` represent the number of inches in the height

Height . . . . . . . . . . . . . . . .`h` `larr` number of inches in the height
2 times that . . . . . . . . . .`2h`
10 more than that . . . `(2h + 10)` `larr` number of inches in the base

Sub those amounts into the general formula

`A = (b*h)/ 2`

`84 = ((2h + 10)(h)) /(2)`

Solve for `h`, already defined as "the number of inches in the height"

`84 = ((2h + 10)(h)) /(2)`

1) Clear the fraction by reducing it to lowest terms
`84 = (h + 5)(h)`

2) Clear the parentheses by distributing the `h`
`84 = h^2 + 5h`

3) Subtract `84` from both sides to set the equation equal to `0`
`h^2 + 5h - 84 = 0`

4) Factor
`(h−7)(h+12)`
https://www.mathpapa.com/factoring-calculator/

5) Set the factors equal to `0` and solve for `h`, already defined as "the number of inches in the height"

`h = - 12` `larr` discarded solution because height cannot be a negative number

`h = 7` `larr` answer for "the number of inches in the height"

`color(white)(..................)`. . . . . . . . . . . .

Find the number of inches in the base

The number of inches in the base is defined as
`2  (h) + 10`
`2 (7) + 10`
`color(white)(.)14   + 10`
`color(white)(......)` `24` `larr` answer for "the number of inches in the base"

`color(white)(..................)`. . . . . . . . . . . .

Answer:
`"Height" = 7 "inches"`
`"Base" = 24 "inches"`

`color(white)(..................)`. . . . . . . . . . . .

Check

Sub the values into the formula and see if the Area still comes out to `84`

`A = (1)/(2)(b*h)`

`A = (1)/(2)(24*7)`

`A = 12*7`

`A = 84`

`Check!`