Question

If the length of each median of an equilateral triangle is 4 cm and then the perimeter of the triangle is:
\[\left( a \right)7\sqrt{3}cm\]
\[\left( b \right)8\sqrt{3}cm\]
\[\left( c \right)9\sqrt{3}cm\]
\[\left( d \right)10\sqrt{3}cm\]

Answer 1

Hint: We are given that the median of an equilateral triangle is 4 cm. First, we will find the side of the equilateral triangle using the Pythagoras theorem which says \[{{P}^{2}}+{{B}^{2}}={{H}^{2}}.\] We will assume the side of the triangle as ‘a’. Then using the above Pythagoras theorem, we get ‘a’ as \[\dfrac{8}{\sqrt{3}}cm.\] Once we have the side, we find the perimeter of the triangle using the formula of the perimeter which is 3 x sides and hence we get our solution.

Complete step-by-step solution:
We are given that the median of an equilateral triangle is 4 cm. We are asked to find the perimeter. The perimeter is defined as the sum of the total length of the boundary. In the triangle, the perimeter is defined as the sum of all the sides. As we have no side given, we will have to find the side of our equilateral triangle.
Let our equilateral triangle be ‘a’ cm. Remember that the median of the equilateral triangle is the perpendicular bisector. In an equilateral triangle ABC with side ‘a’ cm, AD is the median.

As AD is the median, so,
\[CD=DB=\dfrac{BC}{2}\]
As, BC = a, so,
\[\Rightarrow CD=\dfrac{a}{2}\]
We have the median AD as 4 cm. Now in the right triangle ADC, we will apply the Pythagoras theorem.
\[A{{C}^{2}}=A{{D}^{2}}+C{{D}^{2}}\]
As, AC = a, \[CD=\dfrac{a}{2}\] and AD = 4. So,
\[{{a}^{2}}={{4}^{2}}+\dfrac{{{a}^{2}}}{4}\]
\[\Rightarrow {{a}^{2}}-\dfrac{{{a}^{2}}}{4}={{4}^{2}}\]
Simplifying, we get,
\[\Rightarrow \dfrac{4{{a}^{2}}-{{a}^{2}}}{4}={{4}^{2}}\]
\[\Rightarrow \dfrac{3{{a}^{2}}}{4}={{4}^{2}}\]
Now, solving for ‘a’, we get,
\[\Rightarrow a=\dfrac{8}{\sqrt{3}}\]
Now, we have got the side of the triangle as \[a=\dfrac{8}{\sqrt{3}}cm.\]
Now, as in the equilateral triangle, all the sides are equal. So, the perimeter of the equilateral triangle = 3 x side.
\[\text{Perimeter of Equilateral Triangle}=3\times a\]
As, \[a=\dfrac{8}{\sqrt{3}}cm,\] we get,
\[\Rightarrow \text{Perimeter of Equilateral Triangle}=3\times \dfrac{8}{\sqrt{3}}\]
Simplifying further, we get,
\[\Rightarrow \text{Perimeter of Equilateral Triangle}=8\sqrt{3}cm\]
Hence, the correct option is (b).

Note: The alternate method is as follows. We have a median as 4cm. We know that the median of the equilateral triangle is given as \[\dfrac{\sqrt{3}}{2}\times \left( \text{side} \right).\] So, we have,
\[\Rightarrow 4=\dfrac{\sqrt{3}}{2}\times side\]
After solving, we get,
\[\text{side}=\dfrac{8}{\sqrt{3}}\]
Now, the perimeter of the equilateral triangle is given as \[3\times side\] as side is \[\dfrac{8}{\sqrt{3}}cm.\]
So, the perimeter of the equilateral triangle \[=3\times \dfrac{8}{\sqrt{3}}\]
After simplification, we get, the perimeter of the equilateral triangle as \[8\sqrt{3}cm.\]