Question

Hello Mates !!

Class 9

Maths

Prove the :

MID POINT THEORAM

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Answer 1

$<b>Heya mate. (^_-). Solution below.$
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$\red{\bold{\star{\mathcal{What \: is \: the \: midpoint\: Theorem?}}}}$

$<u>==> The midpoint Theorem states that in a triangle, the line segment joining the midpoints of any two sides is parallel to the third side and equal to half of it.</u>$

For the proof, please refer to the attachment.

$<marquee>Hope it helps you.</marquee>$
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Thank you.. ;-)

Answer 2

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 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)

⇒ 1 /2 PR = 1/ 2 BC

⇒ PQ = 1/ 2 BC. [from (4