Ex.6.5 Q16 Triangles Solution - NCERT Maths Class 10

Go back to  'Ex.6.5'

Question

In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.

Diagram

We have to prove \(3B{{C}^{2}}=4A{{D}^{2}}\)

 

Text Solution

Reasoning :

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Steps:

In \(\Delta ABC\)

\(AB = BC = CA\)

\(\begin{align}AD\bot BC\Rightarrow BD=CD=\frac{BC}{2}\end{align}\)

Now In \(\Delta ADC\)

\[\begin{align}&{A C^{2}=A D^{2}+C D^{2}} \\ &{B C^{2}=A D^{2}+\left(\frac{B C}{2}\right)^{2}\left[A C=B C \operatorname{and} C D=\frac{B C}{2}\right]} \\& {B C^{2}=A D^{2}+\frac{B C^{2}}{4}} \\& {B C^{2}-\frac{B C^{2}}{4}=A D^{2}} \\ &{\frac{3 B C^{2}}{4}=A D^{2}} \\ &{3 B C^{2}=4 A D^{2}}\end{align}\]