Here we will prove that converse of the Midpoint Theorem by using the Equal Intercepts Theorem.
Solution:
Given: P is the midpoint of XY in ∆XYZ. PQ ∥ YZ.
To prove: XQ = QZ.
Construction: Through X, draw MN ∥ YZ.
Proof:
Statement |
Reason |
1. PQ ∥ YZ. |
1. MN ∥ YZ and PQ ∥ YZ. |
2. MN ∥ PQ ∥ YZ. |
2. XP = PY. |
3. The transversal XZ also makes equal intercepts XQ and QZ on MN, PQ and YZ. |
3. By the Equal Intercepts Theorem. |
4. Therefore, XQ = QZ. (Proved) |
4. From statement 3. |
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