Overview
The midpoint theorem describes the relationship between a line segment connecting two midpoints of a triangle's sides and the third side, establishing both parallelism and proportional length.
The Midpoint Theorem
- When you connect the midpoints of two sides of a triangle, the resulting segment is parallel to the third side.
- The length of the connecting segment is exactly half the length of the third side.
- Requires two midpoints to be identified on two different sides of the triangle.
- Example: If D is the midpoint of AB and E is the midpoint of AC, then DE is parallel to BC.
- The relationship can be written as: DE = ½ BC.
Applying the Theorem vs Its Converse
| Situation | Given Conditions | Conclusion | Reasoning |
|---|
| Standard Theorem | Two midpoints on triangle sides | Line connecting them is parallel to third side; length is half | Midpoint Theorem (midpt. thm.) |
| Standard Theorem | Two midpoints on triangle sides | Length = ½ of third side | Midpoint Theorem |
| Converse | One midpoint and parallel lines | The parallel line passes through midpoint of opposite side | Converse Midpoint Theorem OR Line through midpoint parallel to other side |
Standard Midpoint Theorem Application
- Start by locating the midpoint of two sides of the triangle (e.g., points C and D).
- Connect the two midpoints with a line segment.
- The new segment CD is automatically parallel to the third side BC.
- The length CD equals exactly half of BC.
- Mathematical statement: CD ∥ BC and CD = ½ BC.
Converse (Reverse) of the Midpoint Theorem
- If one midpoint is given and a line through it is parallel to another side, you can conclude the line passes through the midpoint of the opposite side.
- Example: If E is the midpoint of one side and a line through E is parallel to the base, then the line must pass through the midpoint of the opposite side.
- This proves equal segments on the opposite side without initially knowing both midpoints.
- Teachers may call this "converse midpoint theorem" or "line through midpoint parallel to other side."
- Use this reasoning when given parallelism and one midpoint, but not two midpoints.
Mathematical Notation and Reasoning
- Write CD ∥ BC (CD is parallel to BC); reason: "midpt. thm." (abbreviation acceptable).
- Write CD = ½ BC; reason: "midpoint theorem."
- For converse situations, write AE = EC; reason: "converse midpoint theorem" or "line through midpoint ∥ to other side."
- Always justify conclusions with the appropriate theorem name in mathematical proofs.
Example Problems
- Triangle with BC = 20 cm: If E is midpoint of one side and lines are parallel, then by converse theorem, the opposite side is split into two equal segments of 10 cm each.
- Triangle with BA = 16 cm: Given one midpoint and parallel lines, conclude CD = DB by converse reasoning.
- Triangle where both midpoints are marked: If sides are split equally at two points, DE ∥ BC and DE = 6 cm when BC = 12 cm (midpoint theorem).