Chapter 3.8, Problem 7PSB

To determine

To prove : The altitude of an equilateral triangle is also a median of the triangle.

Explanation of Solution

Given information : The following information has been given

The triangle is equilateral

Let’s assume an equilateral triangle ABC with base BC, and altitude AD

Formula used : All the sides and angles of an equilateral triangle are congruent

Proof : We know that for the triangles ABD and ACD,

  $\angle ADC\cong \angle ADB={90}^0$

AD is the common chord

  $\overrightarrow{AB}\cong \overrightarrow{AC}$ (sides of equilateral triangle)

Thus, we can say that the two triangles are congruent by SAS property, now, we know that congruent triangles have congruent sides. Thus, we can say that

  $\overrightarrow{BD}\cong \overrightarrow{CD}$

This can be said like D is the median of the line BC.

Hence, proved that altitude of an equilateral triangle is also a median of the triangle.