Question

if the length of a median of an equilateral triangle is x cm then find its area

Answer 1

Hey there !

Solution:

In an equilateral triangle, the height and the median are considered to be of same length. Hence we can say that, Height of an equilateral triangle is equal to Median of the same equilateral triangle.

We know that, can be calculated as:

$\text{ Height of an equilateral triangle: } = \dfrac{ \sqrt{3}}{2} a$

Here, 'a' denotes the side of an equilateral triangle.

$\implies x = \dfrac{\sqrt{3}}{2} a \\ \\ \implies a = \dfrac{2x}{ \sqrt{3} }$

Now we know that area of an equilateral triangle can be calculated as:

$\text{ Area of an Equilateral Triangle} = \dfrac{ \sqrt{3} }{4} a^2$

Substituting the value of 'a' we get,

$\implies Area = \dfrac{ \sqrt{3}}{4} \times \dfrac{2x}{ \sqrt{3}} \times \dfrac{ 2x}{\sqrt{3} } \\ \\ \implies \text{ Area} = \dfrac{ \sqrt{3} }{4} \times \dfrac{ 4x^2}{3} \\ \\ \implies \text{ Area } = \dfrac{x^2 \sqrt {3}}{3}$

Which is the required answer !

Hope my answer helped !

Answer 2

Answer:

Concept:
The medians of an equilateral triangle have identical lengths. In the case of an isosceles triangle, the medians from the vertices with equal angles are of the same length.

Step-by-step explanation:

Given:

The length of a median of an equilateral triangle is x cm.

Find:

Find its area
Solution:

Given the Median of the equilateral triangle =X cm.

We know that in an equilateral triangle, the median is also the  altitude (i.e., height) of the triangle

Let the side of the triangle be s.

                    $So, \ BD=S/2 \mathrm{~cm} \\\\In \ right \triangle A D B , A B^{2}=B D^{2}+A D^{2} \\\\ \Rightarrow s^{2}=s^{2} / 4+x^{2} \\\\\Rightarrow x^{2}=s^{2}-s^{2} / 4 \\\\ \Rightarrow \quad x^{2}=3 s^{2} / 4 \\\\or x=\sqrt{3} \mathrm{~s} / 2 or, s=\frac{2 x}{\sqrt{3}}$

                   $or, S=\frac{2 x}{\sqrt{3}} \\\\area of \triangle A B C=\frac{1}{2} \times \frac{2 x}{\sqrt{3}} \times x=\frac{x^{2}}{\sqrt{3}} \mathrm{~cm}^{2}$

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