MidPoint Theorem: Statement, Proof, Definition, Examples

Students need a thorough understanding of a few important geometric concepts to ace their class 8, 9 and 10 exams. The Mid-Point Theorem is one such important theorem in geometry. It states that “the line segment joining the mid-points of two sides of a triangle is parallel to the third side”. This theorem is one of the most important topics from the exam point of view as well. Many questions are framed which require direct application of the mid-point theorem to get solved. So, in this article, we will provide you with all the information on the theorem such as its proof, definition and sample questions as well.

Mid Point Theorem: Definition

As per the definition, the mid-point theorem states that “The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side“.

Mid-point Theorem Proof

To prove the theorem follow the steps mentioned below:

-1st Step: Draw a triangle as given in Fig: 1.
-2nd Step: Join the points E and F.
-3rd Step: Now measure BC and EF.
-4th Step: Measure ∠ ABC & ∠ AEF.
-5th Step: The results will be EF = 1/2 BC and ∠ AEF = ∠ ABC.

Hence proved that “EF || BC“.

In Fig: 2 E and F are the mid-points of AC and AB respectively and BA || CD. So by (ASA Rule) Δ AEF ≅ Δ CDF hence, EF = DF and BE = AE = DC. Therefore, BCDE is a parallelogram.

That gives EF || BC and EF = 1/2 ED = 1/2 BC.

MPT Examples

Example 1: In Δ ABC, D, E and F are respectively the mid-points of sides AB, BC and CA. Show that Δ ABC is divided into four congruent triangles by joining D, E and F.

Solution: As D and E are mid-points of sides AB and BC of the triangle ABC, by mid-point theorem, DE || AC Similarly, DF || BC and EF || AB. Therefore ADEF, BDFE and DFCE are all parallelograms.

Now DE is a diagonal of the parallelogram BDFE, therefore, Δ BDE ≅ Δ FED Similarly Δ DAF ≅ Δ FED and Δ EFC ≅ Δ FED.

Mid-point Theorem Formula

You might be wondering about the formula for the mid-point theorem that you studied in Coordinate Geometry. This formula is applicable for lines and is used to determine the midpoint between two given set of points.

The questions can be framed where you might be given one point and mid-point and you have to find the other point. Another possibility is where both points will be given and you have to find the midpoint.

Let P₁(x₁, y₁) and P₂(x₂, y₂) be the coordinates of 2 given end-points, then the midpoint formula for these points is:

Midpoint = [(x₁+ x₂)/2, (y₁+ y₂)/2]

Sample Questions On Mid Point Theorem

Here are some questions that might be asked in your exams.

1. ABCD is a rhombus and P, Q, R and S are ©wthe mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rectangle.

2. Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

3. ABCD is a rectangle and P, Q, R and S are mid-points of the sides AB, BC, CD and DA respectively. Show that the quadrilateral PQRS is a rhombus.

4. ABCD is a trapezium in which AB || DC, BD is a diagonal and E is the mid-point of AD. A line is drawn through E parallel to AB intersecting BC at F. Show that F is the mid-point of BC

5. In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively. Show that the line segments AF and EC trisect the diagonal BD.

Learn On Embibe

Students can access the following study materials on Embibe for their exam preparation:

NCERT Solutions	NCERT Books
Class 8 Mock Test Series	Class 8 Practice Questions
Class 9 Mock Test Series	Class 9 Practice Questions
Class 10 Mock Test Series	Class 10 Practice Questions
JEE Main Mock Tests (Class 11-12 PCM)	JEE Main Practice Questions (Class 11-12 PCM)
NEET Mock Tests (Class 11-12 PCB)	NEET Practice Questions (Class 11-12 PCB)

FAQs – Frequently Asked Questions

Here are some questions that candidates generally ask:

Q. State the Mid-point theorem.
Ans. The line segment joining the mid-point of two sides of a triangle is parallel to the third side is the statement for the mid-point theorem.

Q. How do you prove the converse of the midpoint theorem?
Ans. The proof for the converse of the theorem with the figure has been provided on this page.

Q. What is the midpoint formula?
Ans. The mid-point formula is [(x₁+ x₂)/2, (y₁+ y₂)/2].

So after providing you with all the necessary information on the mid-point theorem we have reached the end of our article. Hope you enjoyed learning, however, if you have any more questions use the comments section and we will surely update you.