Ex.6.5 Q6 Triangles Solution - NCERT Maths Class 10

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


Text Solution


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.


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

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