Question

If all the three altitudes of a triangle are equal, the triangle is equilateral.
State whether the above statement is true or false.

• A. True
• B. False

Answer 1

The correct option is A True

Consider triangles BEC and BFC
$EC=BF$ (Equal altitudes: Given)
$\angle BEC=\angle BFC={90}^o$ (BE and BF are Altitudes)
BC = BC (common)
$△BEC\cong △BFC$ (RHS postulate)
$\angle ABC=\angle BCA$ (Corresponding angles)
Consider the $△$ ADB and $△$ ADC
$\angle ADB=\angle ADC={90}^o$ (AD is Altitude)
$AD=AD$ (common)
$\angle ABC=\angle BCA$
Thus, $△ADB\cong △ADC$ (ASA postulate)
$AB=AC$ (Corresponding sides are equal)
$BC=AC$ (Similarly, we can prove $△BFC\cong △BFA$)
Thus, $AB=AC=BC$