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Define mid point theorom |
| Answer» 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.Given: In triangle ABC, P and Q are mid-points of AB and AC respectively.To Prove: i) PQ || BC ii) PQ = 1/ 2 BCConstruction: 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 = CRAlso. BP || CR. (by construction)In quadrilateral BCRP, BP = CR and BP || CRTherefore, quadrilateral BCRP is a parallelogram.BC || PR or, BC || PQAlso, PR = BC (∵ BCRP is a parallelogram)⇒ 1 /2 PR = 1/ 2 BC⇒ PQ = 1/ 2 BC. [from (4)] | |