Mid point theorem

 Geometry is an important part of mathematics that deals with different shapes and figures. Triangles are an important part of geometry and the mid-point theorem points towards mid points of the triangle.

What is Mid-Point Theorem?

This theorem states that” The line segment joining mid-points of two sides of a triangle is parallel to the third side of the triangle and is half of it”

Proof of Mid-Point Theorem

A triangle ABC in which D is the mid-point of AB and E is the mid-point of AC.

To Prove: DE ∥ BC and DE = 1/2(BC)

Construction

Extend the line segment joining points D and E to F such that DE = EF and join CF.

Proof

In ∆AED and ∆CEF 

DE = EF (construction)

∠1 = ∠2 (vertically opposite angles)

AE = CE (E is the mid-point)

△AED ≅ △CEF by SAS criteria

Therefore, 

∠3 =∠4 (c.p.c.t)

But these are alternate interior angles.

So, AB ∥ CF

AD = CF(c.p.c.t)

But AD = DB (D is the mid-point)

Therefore, BD = CF

In BCFD

BD∥ CF (as AB ∥ CF)

BD = CF

BCFD is a parallelogram as one pair of opposite sides is parallel and equal.

Therefore, 

DF∥ BC (opposite sides of parallelogram)

DF = BC (opposite sides of parallelogram)

As DF∥ BC, DE∥ BC and DF = BC

But DE = EF

So, DF = 2(DE)

2(DE) = BC

DE = 1/2(BC)

Hence, proved that the line joining mid-points of two sides of the triangle is parallel to the third side and is half of it.

What is the Converse of Mid-Point Theorem?

The line drawn through the mid-point of one side of a triangle parallel to the base of a triangle bisects the third side of the triangle.

Proof of the Theorem

In triangle PQR, S is the mid-point of PQ and ST ∥ QR 

To Prove: T is the mid-point of PR.

Construction

Draw a line through R parallel to PQ and extend ST to U.

Proof

ST∥ QR(given)

So, SU∥ QR

PQ∥ RU (construction)

Therefore, SURQ is a parallelogram.

SQ = RU(Opposite sides of parallelogram)

But SQ = PS (S is the mid-point of PQ)

Therefore, RU = PS

In △PST and △RUT

∠1 =∠2(vertically opposite angles)

∠3 =∠4(alternate angles)

PS = RU(proved above)

△PST ≅ △RUT by AAS criteria

Therefore, PT = RT

T is the mid-point of PR.