Question

If the perpendicular bisector of one side of a triangle goes through the opposite vertex, then is the triangle isosceles?

Answer 1

True

Explanation:

The perpendicular bisector of the opposite side to the vertex bisects the angle at the vertex into two equal parts and also bisects the triangle into two equal parts.

Let A be the angle at the vertex, then assume that the angle is an isosceles triangle with base angles B.

We need to show that A = 180 - 2B for an isoceles triangle

The perpendicular bisector bisects A into two so the new angle in the vertex one half of the bisected triangle is A/2.

Since this half triangle is a right-angled triangle, the third angle in it is 90.

So, A/2 + B + 90 = 180 (Sum of angles in a triangle)

subtracting 90 from both sides, we have

A/2 + B + 90 - 90 = 180 - 90

A/2 + B = 90

subtracting B from both sides, we have

A/2 + B = 90

A/2 = 90 - B

multiplying through by 2, we have

A = 2(90 - B)

A = 180 - 2B

Since A = 180 - 2B, then our triangle is an isosceles triangle.