Question

Calculate the area and the height of an equilateral triangle whose perimeter is 60 cm.
A. $173.2c{{m}^{2}};11.32$
B. $173.2c{{m}^{2}};19.32$
C. $173.2c{{m}^{2}};17.32$
D. $173.2c{{m}^{2}};14.32$

Answer 1

Hint:For solving this problem, first we find the side length by using the perimeter. Now, the height of an equilateral triangle is evaluated by using Pythagoras theorem. After this, by using the formula of area of the triangle, we evaluate the area of the equilateral triangle. In this way we can easily solve the problem.

Complete step-by-step answer:
According to the problem statement, the perimeter of an equilateral triangle is 60cm. Let all the three equal sides of the equilateral triangle be a each.
The formula for the perimeter of a triangle is the sum of all the sides. So, by using the formula of perimeter of a equilateral triangle:
$\begin{align}
  & a+a+a=60 \\
 & 3a=60 \\
 & a=\dfrac{60}{3} \\
 & a=20cm \\
\end{align}$
Therefore, the side length of the equilateral triangle is 20cm.
Now, by using the property that the median of the equilateral triangle bisects the side, calculate the half length of the side.

Applying the Pythagoras theorem in triangle ABD,
$\begin{align}
  & A{{D}^{2}}=A{{B}^{2}}+B{{D}^{2}} \\
 & {{20}^{2}}={{h}^{2}}+{{10}^{2}} \\
 & {{h}^{2}}=400-100 \\
 & {{h}^{2}}=300 \\
 & h=\sqrt{300} \\
 & h=17.32cm \\
\end{align}$
Therefore, the height of the triangle is 17.32 cm.
Now, the area of a triangle is given by $=\dfrac{1}{2}\times b\times h$
On putting the values as b = 20 cm and h = 17.32 cm in area, we get
$\begin{align}
  & A=\dfrac{1}{2}\times 20\times 17.32 \\
 & A=173.2c{{m}^{2}} \\
\end{align}$
Therefore, the area of the triangle is 173.2 square cm.
Hence, option (C) is correct.

Note: This problem can also be solved alternatively by using the formula for or area of equilateral triangle which can be stated as $\dfrac{\sqrt{3}}{4}{{a}^{2}}$, where a is the side of triangle. Once the area is obtained then by using the general formula for area of triangle i.e $=\dfrac{1}{2}\times b\times h$ height can be calculated.