Question

Can somone explain me mid point theorem

Answer 1

I hope it will help you

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 = 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)]

Pls mark as a brainlist

Answer 2

the mid point theorem says that in a triangle, a line running parallel to one side of the triangle and which is also running through mid point of one of the other sides, divides the other side from mid too.
OR its converse:
A line joining the mid points of two sides of a triangle is parallel to the third side and is half of it.