state and prove the converse mid point theorem
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Answer:
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”.
Step-by-step explanation:
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.
MidPoint Theorem Formula
In Coordinate Geometry, midpoint theorem refers to the midpoint of the line segment. It defines the coordinate points of the midpoint of the line segment can be found by taking the average of the coordinates of the given endpoints. The midpoint formula is used to determine the midpoint between the two given points.
If P1(x1, y1) and p2(x2, y2) are the coordinates two given endpoints, then the midpoint formula is given as:
Midpoint = ((x1+x2)/2, (y1+y2)/2)
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.
