stand and prove the converse of mid point theorem
Answers
Answer:
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:
1. SU ∥ QR and SU = 12QR.
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.
4. ST and SU are not the same line.
4. From statement 3.
5. T and U are coincident points.
5. From statement 4.
6. T is the midpoint of PR (Proved).
6. From statement 5.
The converse of MidPoint Theorem
The converse of the midpoint theorem states that ” if a line is drawn through the midpoint of one side of a triangle, and parallel to the other side, it bisects the third side”.
Given: In ∆PQR, S is the midpoint of PQ, and ST is drawn parallel to QR.
Converse of Midpoint Theorem Proof
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To prove: ST bisects PR, i.e., PT = TR.
Construction: Join SU where U is the midpoint of PR.
Converse of Midpoint Theorem
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Proof:
Statement
Reason
1. SU ∥ QR and SU = 12QR.
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.
4. ST and SU are not the same line.
4. From statement 3.
5. T and U are coincident points.
5. From statement 4.
6. T is the midpoint of PR (Proved).
6. From statement 5.
