# The height of a triangle is 4 inches more than twice the length of the base. The area of the triangle is 35 square inches. What is the height of the triangle?

Feb 24, 2016

$14$

#### Explanation:

Let the Base of the $\triangle$ be color(red)(x

Then the Height will be color(red)(2x+4

Area of $\triangle$=color(brown)(1/2bh

Where,

color(brown)(b=base,h=height,Area=35 (in this case)

Substitute the values into the equation

$\rightarrow 35 = \frac{1}{2} \left(2 x + 4\right) \left(x\right)$

$\rightarrow 35 = \frac{\left(\cancel{2} x + \cancel{4}\right)}{\cancel{2}} \left(x\right)$

$\rightarrow 35 = \left(x + 2\right) \left(x\right)$

$\rightarrow 35 = {x}^{2} + 2 x$

Subtract $35$ both sides

$\rightarrow 0 = {x}^{2} + 2 x - 35$

Rewrite the equation in the Standard form

${x}^{2} + 2 x - 35 = 0$

Factor the equation

$\rightarrow \left(x + 7\right) \left(x - 5\right) = 0$

So we have color(blue)(x=-7,5

length or distance should not be $\underline{n}$$\underline{e} \underline{g} \underline{a} \underline{t} \underline{i} \underline{v} \underline{e}$ numbers

So color(orange)(x=5

They have asked us to find the Height

So,

rArrcolor(green)(Height=2x+4=2(5)+4=10+4=14

Feb 24, 2016

height $= 14$ inches.

#### Explanation:

Let the height be $h$ and the base be $h$ (inches)
$h = 2 b + 4$

Area: $\frac{b h}{2} = 35$

$\textcolor{w h i t e}{\text{XXX}} b \times \left(2 b + 4\right) = 70$

$\textcolor{w h i t e}{\text{XXX}} 2 {b}^{2} + 4 b = 70$

$\textcolor{w h i t e}{\text{XXX}} {b}^{2} + 2 b - 35 = 0$

$\textcolor{w h i t e}{\text{XXX}} \left(b - 5\right) \left(b + 7\right) = 0$

$\Rightarrow b = 5 \mathmr{and} b = - 7$

Since the base must be positive:
$\textcolor{w h i t e}{\text{XXX}} b = 5$
and
$\textcolor{w h i t e}{\text{XXX}} h = 2 b + 4 = 14$