This is a theorem about the segment joining the midpoints of two sides of a triangle.
When the midpoint M of side AB and the midpoint N of side AC of △ABC are joined,
MN ∥ BC and MN = BC/2
Example ①
In △ABC, let M be the midpoint of AB and N the midpoint of AC. When BC=12cm, find MN.
Solution: MN = BC/2 = 6 cm
In △ABC, let M be the midpoint of AB and N the midpoint of AC. When BC=12cm, find MN.
Solution: MN = BC/2 = 6 cm
Example ② (Using the Converse)
When MN ∥ BC and MN=7cm, find BC.
Solution: BC = 2×MN = 14 cm
When MN ∥ BC and MN=7cm, find BC.
Solution: BC = 2×MN = 14 cm
Key Point
- Midpoint theorem: MN = 1/2 BC, MN ∥ BC
- Check that the points are “midpoints,” since this is a condition.
Practice Problem
- In △ABC, M and N are the midpoints of AB and AC, respectively. When MN=5cm, find BC.
Answer and Explanation
- Answer: BC=10cm. MN=BC/2 → BC=2MN=10.