Chapter 7.3, Problem 18PSB

To determine

To prove : The Midline theorem

Explanation of Solution

Given information : The following information has been given

A triangle ABO, with AO as base

MN as midline of triangle

Formula used : The Midline theorem states that

The segments that joins the midpoints of two sides of a triangle is parallel to the third side and half as long

Proof : Let’s assume that MN are two midpoints of sides BO and BA respectively. Now, to prove the sides parallel, we prove that the angles are congruent, hence the slopes will be same.

Thus, we get M and N as

  $M=(b,c)$ and

  $N=(a+b,c)$

Now, we need to compare slopes of AO and MN.

Slope of AO $=\frac{0-0}{2a-0}=0$

Slope of MN $=\frac{c-c}{a+b-b}=0$

Hence, we get the slopes as same. Thus, proved that midline theorem exists.