Question

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?

Answer 1

Finding the height of the triangle:

Step 2: Find the measure of $b$

Suppose the length of the base of the triangle is $b$ inches.

Now, given that the height is $4$ inches more than twice the length of the base. So, the length of the height will be $2b+4$ inches.

We know that the area of a triangle whose base length is $b$ and height is $h$ is given by: $A=\frac{1}{2}bh$.

Here, $A=35$ square inches, $h=2b+4$ inches.

So, using the above formula, we get:

$\begin{array}{rcl}A& =& \frac{1}{2}bh\\ & \Rightarrow & 35=\frac{b\left(2b+4\right)}{2}\\ & \Rightarrow & 35=b\left(b+2\right)\\ & \Rightarrow & b^2+2b-35=0\\ & \Rightarrow & b^2+7b-5b-35=0\\ & \Rightarrow & b\left(b+7\right)-5\left(b+7\right)=0\\ & \Rightarrow & \left(b+7\right)\left(b-5\right)=0\end{array}$

Therefore, $b=-7$ or $b=5$. As length cannot be negative, so we must take, $b=5$.

Step 3: Finding the height

Since, $h=2b+4$ and $b=5$, we get:

$$\begin{gathered} h=2b+4 \\ =2\times 5+4 \\ =14 \end{gathered}$$

Therefore, the height of the triangle is $14$ inches.