Converse of Midpoint Theorem
The straight line drawn through the midpoint of one side of a triangle parallel to another bisects the third side.
Given: In ∆PQR, S is the midpoint of PQ, and ST is drawn parallel to QR.
To prove: ST bisects PR, i.e., PT = TR.
Construction: Join SU where U is the midpoint of PR.
Proof:
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Statement |
Reason |
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1. SU ∥ QR and SU = \(\frac{1}{2}\)QR. |
1. By Midpoint Theorem. |
|
2. ST ∥QR and SU ∥ QR. |
2. Given and statement 1. |
|
3. ST ∥ SU. |
3. Two lines parallel to the same line are parallel themselves. |
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4. ST and SU are not the same line. |
4. From statement 3. |
|
5. T and U are coincident points. |
5. From statement 4. |
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6. T is the midpoint of PR (Proved). |
6. From statement 5. |
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