Ex.6.5 Q6 Triangles Solution - NCERT Maths Class 10

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Question

\(ABC\) is an equilateral triangle of side \(2\)\(a.\) Find each of its altitudes.

Diagram

Text Solution

Reasoning:

We know that in an equilateral triangle perpendicular drawn from vertex to the opposite side, bisects the side.

As we know that, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Steps:

In \(\,\,\Delta ABC\)

\[A B=B C=C A=2 a\]

\(A D \perp B C\) 

\[\begin{align}\Rightarrow\; B D&=C D=\frac{{1}}{2}BC={a}\end{align}\]

In \(\Delta ADB\) , 

\[\begin{align} A B^{2} &=A D^{2}+B D^{2} \\  A D^{2} &=A B^{2}-B D^{2} \\ &=(2 a)^{2}-a^{2} \\ &=4 a^{2}-a^{2} \\ &=3 a^{2} \\AD &=\sqrt{3} a\\ \Rightarrow A D &=\sqrt{3} a \text { units } \end{align}\]

Similarly, we can prove that

\[\mathrm{BE}=\mathrm{CF}=\sqrt{3} a \text { units }\]