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Mid Point Theorem

Last Updated : 16 May, 2024
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Midpoint theorem states that “The line segment drawn from the midpoint of two sides of the triangle is parallel to the third side and is equal to half of the third side of the triangle. 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 the midpoints of the triangle.

What is the Midpoint Theorem?

The Midpoint theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and equal to half of the third side. Consider a triangle, ΔABC. Let D and E be the midpoints of AB and AC respectively, now join DE. Then the midpoint theorem states that DE is parallel to BC and equal to half of BC.

Mid Point Theorem Definition

The midpoint theorem, also known as the mid-segment theorem or midpoint formula, states that in a triangle, the segment joining the midpoints of two sides is parallel to the third side and half of its length

Mid-Point Theorem Proof

This theorem states that”

“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”

Mid-Point-Theorem

Given:

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 the mid-points of two sides of the triangle is parallel to the third side and is half of it.

Mid-Point Theorem Converse

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.

Converse of Mid-Point Theorem

Given:

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.

Mid Point Formula

Midpoint of any line segment is defined as the coordinate point that divides the line segment into two equal parts.

Suppose P(x1, y1) and Q(x2, y2) are the coordinates of endpoints of any line segment, then the midpoint formula is given as:

Midpoint = [(x1 + x2)/2, (y1 + y2)/2]

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Examples on Mid Point Theorem

Example 1: l, m, and n are three parallel lines. p and q are two transversals intersecting parallel lines at A, B, C, D, E, and F as shown in the figure. If AB:BC = 1:1, find the ratio of DE:EF.

Example 1 on Mid Point Theorem

Given: AB:BC = 1:1

To find: DE:EF

Construction: Join AF such that it intersects line m at G.

In △ACF 

AB = BC (1:1 ratio)

BG∥ CF (as m∥n)

Therefore, by converse of mid-point theorem G is the midpoint of AF (AG = GF)

Now, in △AFD

AG = GF (proved above)

GE∥ AD (as l∥m)

Therefore, by converse of mid-point theorem E is the mid-point of DF (FE = DE)

So, DE:EF = 1:1 (as they are equal)

Example 2: In the figure given below L, M and N are mid-points of side PQ, QR, and PR respectively of triangle PQR.

If PQ = 8cm, QR = 9cm and PR = 6cm. Find the perimeter of the triangle formed by joining L, M, and N.

Example 2 on Mid Point Theorem

Solution:

As L and N are mid-points

By mid-point theorem 

LN ∥ QR and LN = 1/2 × (QR)

LN = 1/2 × 9 = 4.5cm

Similarly, LM = 1/2 × (PR) = 1/2×(6) = 3cm

Similarly, MN = 1/2 × (PQ) = 1/2 × (8) = 4cm

Therefore,

Perimeter of △LMN = LM + MN + LN

Perimeter of △LMN = 3 + 4 + 4.5 = 11.5 cm

Perimeter is 11.5cm

FAQs on Mid Point Theorem

What is Meant by Midpoint Theorem?

Midpoint theorem states that “The line joining the midpoint of two sides of the triangle is parallel to its third side of the triangle and is also half of it.”

What is Converse of Midpoint Theorem?

Converse of mid point theorem states that,” The line drawn through the midpoint of any one side of a triangle, and parallel to another side bisects the third side.

What is Definition of a Midpoint?

Midpoint is defined as the point that divides any line segment into two equal parts.

What is Mid Point Theorem Class 9?

The midpoint theorem states that “The line segment in a triangle joining the midpoint of any two sides of the triangle is said to be parallel to its third side and is also half of the length of the third side.”



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