prove the converse of the mid point theorem
Answers
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:
Statement
Reason
1. SU ∥ QR and SU = 12QR.
By Midpoint Theorem.
2. ST ∥QR and SU ∥ QR.
Given and statement 1.
3.ST ∥ SU.
Two lines parallel to the same line are parallel themselves.
4. ST and SU are not the same line.
From statement 3.
5. T and U are coincident points.
From statement 4.
6. T is the midpoint of PR (Proved).
From statement 5.
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Mid point Theorem :
The line segment joining the mid points of any two sides of a triangle is parallel to the third side.
Given :
A \triangle ABC△ABC in which D and E are the mid points of AB and AC, respectively.
To prove :
DE \parallel BCDE∥BC.
Proof :
Since D and E are the mid points of AB and AC, respectively, we have AD=DBAD=DB and AE=ECAE=EC.
Therefore,
\dfrac{AD}{DB}=\dfrac{AE}{EC}
DB
AD
=
EC
AE
( each equal to 1 )
Therefore, by the converse of thales theorem, DE \parallel BCDE∥BC.
