Question

If the three altitudes of a triangle are equal, then the triangle is

• A. isosceles
• B. right angled triangle
• C. equilateral
• D. none of these

Answer 1

The correct option is C equilateral

Let $AD,BE$ and $CF$ are the altitudes drawn on sides $BC,CA$ and $AB$ of such that $AD=BE=CF.$

Since area of triangle is $\frac{1}{2}$ base $\times$ altitude.

Therefore, the area of is given by $:$

$\frac{1}{2}\times BC\times AD=\frac{1}{2}\times AB\times CF=\frac{1}{2}\times CA\times BE$

$\Rightarrow BC\times AD=AB\times CF=CA\times BE$

$\Rightarrow$$BC=AB=CA$ ( as $AD=BE=CF$)

Thus is an equilateral triangle.

Hence, option $C$ is correct.