What is mid point theorem ?Explain!
Answers
Answer:
The Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.
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fig. mid pt theorem
Mid point theorem :-
"The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the third side."
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Given: In triangle ABC, P and Q are mid-points of AB and AC respectively.
To Prove: i) PQ || BC
ii) PQ = 1/ 2 BC
Construction: Draw CR || BA to meet PQ produced at R.
Proof:
∠QAP = ∠QCR. (Pair of alternate angles) ---------- (1)
AQ = QC. (∵ Q is the mid-point of side AC) ---------- (2)
∠AQP = ∠CQR (Vertically opposite angles) ---------- (3)
Thus, ΔAPQ ≅ ΔCRQ (ASA Congruence rule)
PQ = QR. (by CPCT). or PQ = 1/ 2 PR ---------- (4)
⇒ AP = CR (by CPCT) ........(5)
But, AP = BP. (∵ P is the mid-point of the side AB)
⇒ BP = CR
Also. BP || CR. (by construction)
In quadrilateral BCRP, BP = CR and BP || CR
Therefore, quadrilateral BCRP is a parallelogram.
BC || PR or, BC || PQ
Also, PR = BC (∵ BCRP is a parallelogram)
⇒ PQ = 1/ 2 BC [from (4)]
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