Question

In a triangle, if the altitude (the perpendicular line from a vertex to the opposite side) is also a median (a line segment from a vertex to the midpoint of the opposite side), then is the triangle equilateral?

A) Always
B) Sometimes
C) Never

Answer 1

If the altitude of a triangle is also a median, then the triangle is sometimes equilateral. An equilateral triangle is a triangle with all three sides and angles equal.

Step-by-step explanation:

If the altitude of a triangle is also a median, then the triangle is sometimes equilateral. An equilateral triangle is a triangle with all three sides and angles equal. Let's consider a triangle with altitude AD and median AE as shown below:

If the altitude AD is also a median, then it must divide the base BC into two equal parts. Let's say BD = DC = x. Since the triangle is equilateral, all sides are equal, so AB = BC = AC. Thus, we have the equation AB = x + x = 2x. Since AD is perpendicular to BC, we have the equation AD + BD + DC = AB, which simplifies to AD + x + x = 2x. Solving for AD, we get AD = x.

Therefore, in an equilateral triangle, all three altitudes are also medians. However, in a triangle where the altitude is also a median, it is not necessary for the triangle to be equilateral, as there can be non-equilateral triangles with an altitude that is also a median.