prove mid- point theorem?
Answers
Answer:
Given in the figure A :
AP=PB, AQ=QC.
To prove:
PQ || BC and PQ=1/2 BC
Plan:
To prove ▲ APQ ≅ ▲ QRC
Proof steps:
AQ=QC [midpoint]
∠ APQ = ∠QRC [Corresponding angles for parallel lines cut by an transversal].
∠PBR=∠QRC=∠APQ [Corresponding angles for parallel lines cut by an transversal].
∠RQC=∠PAQ [When 2 pairs of corresponding angles are congruent in a triangle, the third pair is also congruent.]
Therefore ,
▲APQ ≅ ▲QRC
AP=QR=PB and PQ=BR=RC.
Since midpoints are unique, and the lines connecting points are unique, the proposition is proven.
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.
