Mid point theorm of class 9th CBSE board
Answers
The mid-point of the line segment is the geometric centre of the line segment. Mid-point of a line segment divides it into two equal halves.
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 = 12 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 = 12 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)
⇒ 12 PR = 12 BC
⇒ PQ = 12 BC [from (4)]
Converse of Mid-Point Theorem
The line drawn through the mid-point of one side of a triangle and parallel to another side bisects the third side.
midpoint, midpoint theorem, parallel to third side, half the third side
In triangle ABC, if P is the mid-point of AB and PQ || BC then, AQ = QC.
Quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order is a parallelogram.