Question

One side of an equilateral triangle measures \[8{\text{ }}cm.\] Find its area using Heron’s formula. What is its altitude?
A. $ 12\sqrt 3 c{m^2},3\sqrt 3 cm $
B. $ 16\sqrt 3 c{m^2},4\sqrt 3 cm $
C. $ 14\sqrt 3 c{m^2},\dfrac{7}{2}\sqrt 3 cm $
D. $ 20\sqrt 3 c{m^2},5\sqrt 3 cm $

Answer 1

Hint: To solve this question, we will construct a figure of an equilateral triangle to understand better, and then first by applying the formula of Heron’s formula, we will get the value of area of triangle, and then by using the general formula of area of triangle, and putting all the values in it, we will get the value of altitude of the triangle.

Complete step-by-step answer:
We have been given that an equilateral triangle has side equals to \[8{\text{ }}cm,\] we need to find its area and also its altitude.
So, at first let us draw a figure, to understand better.

Here in the figure, AB \[ = \] BC \[ = \] AC.
We know that, according to Heron’s formula, if a triangle has side equals to a, b and c, then the area of triangle, $ \Rightarrow A = \sqrt {s(s - a)(s - b)(s - c)} $
where, $\Rightarrow s = \dfrac{{a + b + c}}{2} $
Since, we have been given an equilateral triangle, then \[a = b = c = 8cm.\] So, on applying the values in the above formula, we get
 $
\Rightarrow s = \dfrac{{8 + 8 + 8}}{2} \\
\Rightarrow s = \dfrac{{24}}{2} \\
\Rightarrow s = 12cm \\
  $
Area of triangle, $ A = \sqrt {12(12 - 8)(12 - 8)(12 - 8)} $
 $
\Rightarrow A = \sqrt {(3 \times 4) \times (4) \times (4) \times (4)} \\
\Rightarrow A = \sqrt {(3 \times 4) \times (4) \times (4) \times (4)} \\
\Rightarrow A = 16\sqrt 3 c{m^2} \\
 $
Now, there is another formula of area of triangle, i.e., $ A = \dfrac{1}{2} \times base \times altitude $
So, we have area of triangle $ = 16\sqrt 3 c{m^2} $
Base of triangle \[ = {\text{ }}8cm\]
On putting the values in the above formula, we get
 $ 16\sqrt 3 = \dfrac{1}{2} \times 8 \times Altitude $
Altitude $ = \dfrac{{16\sqrt 3 }}{4} $ $ = 4\sqrt 3 cm $
Thus, the area of the triangle is $ 16\sqrt 3 c{m^2} $ and its altitude is $ 4\sqrt 3 cm. $

So, the correct answer is “Option B”.

Note: Here, we are asked to find the area of an equilateral triangle by using heron's formula or also called as heron’s formula, which is, $ A = \sqrt {s(s - a)(s - b)(s - c)} $ where, $ s = \dfrac{{a + b + c}}{2} $
In the formula, s is the semi perimeter of the triangle. So, at first, we need to find the value of the semi perimeter of the triangle, then on applying its value in the formula, we will get the area of the triangle.