Question

using theorem 6.2, prove that the line joining the mid-points of any two sides of a triangle is parallel to the third side.

Answer 1

Using Theorem 6.2, it can be proven that the line joining the midpoints of any two sides of a triangle is parallel to the third side.

Step-by-step explanation:

The proof of this statement can be done using Theorem 6.2, which states that if a line is drawn parallel to one side of a triangle and intersecting the other two sides, then it divides those two sides proportionally.

To prove that the line joining the midpoints of any two sides of a triangle is parallel to the third side, we can use this theorem.

Let's consider a triangle ABC with sides AB, BC, and AC. Let M and N be the midpoints of sides AB and AC, respectively. We want to prove that MN is parallel to BC.

1. Using Theorem 6.2, draw a line through M parallel to AC, and let it intersect BC at point P.
2. Since MN is parallel to AC, and MP is parallel to AC, we have that MN is parallel to MP.
3. Using Theorem 6.2 again, we can conclude that MP divides AB in proportional segments, meaning that AP is half of BC.
4. But since MN is parallel to MP, MN is also half of BC. This implies that MN is parallel to BC, as desired.

Therefore, we have proven that the line joining the midpoints of any two sides of a triangle is parallel to the third side.