Question

If two altitudes of a triangle are equal in length, then the triangle is

• A. right angled
• B. equilateral
• C. isosceles
• D. scalene

Answer 1

Answer: (C)

isosceles

Let $ABC$ be a triangle with altitudes $AD$ and $BE$ of equal length as shown in figure.

Consider the triangles $ADC$ and $BEC$.

They are the right triangles with the common angle $ACB$.

The angles $CAD$ and $CBE$ are congruent as the complementary angles to the angle $ACB$.

Thus, the triangles $ADC$ and $BEC$ have congruent sides $AD$ and $B$E as well as congruent angles $ADC$ and $BEC$ (right angle) and congruent angles $CAD$ and $CBE$.

Therefore, the triangles $ADC$ and $BEC$ are congruent, in accordance to the postulate P2 (ASA).

Thus, the straight segments $AC$ and $BC$ are of equal length as the corresponding sides of these triangles and in other words, the triangle is isosceles.

Hence, option C is correct.