Question

How do you find the perimeter of a triangle if the altitude of an equilateral triangle is $32cm$ ?

Answer 1

Hint: Problems of finding the perimeter of an equilateral triangle can be done with the help of a simple trigonometric formula of triangles when the altitude of the triangle is given. The formula will give us the relationship between the altitude of an equilateral triangle with the length of its one side and from that we will have the length of the side of the triangle. Eventually we will get the value of the perimeter of the triangle by using the length of the side.

Complete step-by-step solution:
Let's take the length of one side of an equilateral triangle to be $l$ and the altitude to be $h$.

Now, using a simple trigonometric function we get a relationship between the altitude and length of one side of the triangle as shown below
Altitude of the triangle, $h=l\sin {{60}^{\circ }}$
Now, putting the value of $\sin {{60}^{\circ }}$ in the above relationship we get
$\Rightarrow h=l\times \dfrac{\sqrt{3}}{2}$
Cross multiplying the terms in the denominator of the above expression we get
$\Rightarrow l=\dfrac{2h}{\sqrt{3}}$
We now substitute the given value of the altitude in the above relationship established between the length of one side of equilateral triangle and its altitude
$\Rightarrow l=\dfrac{2\times 32}{\sqrt{3}}cm$
Further simplifying the above expression, we get
$\Rightarrow l=36.95cm$
We know that the perimeter of an equilateral triangle is thrice the length of one side of the same equilateral triangle
Hence,
$Perimeter=3\times l$
$\begin{align}
  & \Rightarrow Perimeter=3\times 36.95cm \\
 & \Rightarrow Perimeter=110.85cm \\
\end{align}$
Therefore, we conclude that the perimeter of the equilateral triangle is $110.85cm$.

Note: We should carefully note the mention of altitude in the given question and should not blindly take the length of the equilateral triangle to be $32cm$ as doing so will lead to wrong answers. Altitude of an equilateral triangle can also be expressed as $l\cos {{30}^{\circ }}$ by taking the semi-angle. The formula of perimeter in terms of altitude can be memorised for easy solution, which is $Perimeter=2\sqrt{3}h$