Ex.6.5 Q16 Triangles Solution - NCERT Maths Class 10

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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.


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.


In \(\Delta ABC\)

\(AB = BC = CA\)

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

Now In \(\Delta ADC\)

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