Question

The base of a triangle is x feet and its height is 3 more than twice the base. If the area of the triangle is 126 square feet, what are the dimensions of the triangle?

Answer 1

Base = `21/2`

Height = `24`

Explanation:

Start with the formula for the area

`A = \frac{Bh}{2}`

where `B` is the base and `h` is the height. We know that `B=x` and `h=2B+3 = 2x+3`. Plug these values into the formula to get

`A = \frac{x(2x+3)}{2} = \frac{2x^2+3x}{2}`

Now we can plug the value for the area as well:

`\frac{2x^2+3x}{2}=126`

and multiply both sides by `2`:

`2x^2+3x = 252`

subtract `252` from both sides:

`2x^2+3x-252=0`

this is a quadratic equation, like `ax^2+bx+c=0`. You can solve these equations with the quadratic formula to find the solutions

`x_{1,2} = \frac{-b\pm\sqrt{b^2-4ac}}{2a}`

In your case,

`x_{1,2} = \frac{-3\pm\sqrt{9+2016}}{4} = \frac{-3\pm\sqrt{2025}}{4} = \frac{-3\pm45}{4}`

So, one solution is

`\frac{-3+45}{4}= 42/4=21/2`

and the other is

`\frac{-3-45}{4}= -48/4=-12`

These solution are algebraically valid, but we come from a geometry problem, where negative lengths have no meaning. So, we reject this last solution.

So, we have `B=x=21/2` for the base, while for the height we have

`h = 2x+3 = 2*21/2+3= 21+3=24`

Answer 2

`color(blue)(a=(3sqrt(1073))/4, b=(3sqrt(1073))/4, c=21/2" feet")`

Explanation:

Base of triangle is `x`

Height is 3 more than twice the base.

`"height=2x+3`

Area of triangle is:

`"Area"=1/2"base"*"height"`

`"Area"=1/2x*(2x+3)=126`

Solving for `x`

`x*(2x+3)=252`

`2x^2+3x-252=0`

Factor:

`(x+12)(2x-21)=0=>x=21/2 and x=-12`

`x=-12` not valid for this problem. ( negative length )

So we have:

`"Base"=21/2`

`"Height"=2(21/2)+3=24`

If we assume the triangle to be isosceles and denote the two equal sides as `bba` and `bb(b)`, then using Pythagoras' theorem:

`a=sqrt((1/2"base")^2+("height")^2`

`a=sqrt((21/4)^2+(24)^2)=(3sqrt(1073))/4`

`b=a=(3sqrt(1073))/4`

Dimensions of triangle are:

`a=(3sqrt(1073))/4, b=(3sqrt(1073))/4, c=21/2`feet

Notice that the type of triangle is not unique in this problem. We could have chosen a right angled triangle instead:

`a=24, b=21/2`

By Pythagoras' theorem:

`c=sqrt((24)^2+(21/2)^2)=(3sqrt(1073))/4`

`"Area"=1/2(21/2)*24=126`