**MidPoint Theorem Complete Explanation**

**Midpoint theorem** is compulsory in maths subject class 9 of each board wether it is CBSE or any other state boards. The questions based on **mid point theorem** are asked every year in CBSE class 9 half-yearly and annual exams. Here in our website **future study point,** you can study **NCERT solutions of maths and science**, most important **questions of maths and science**, **sample papers, guess papers, useful articles on** **science and maths** for your preparation of SA1 and SA2 exams**,** articles on your carrier and other educational posts and topics related to the carrier in online jobs. If you like this post then please fill up our membership form so that you could get updated information on your mobile regarding our new posts. Here in this post, the verification of the **midpoint theorem** is completely explained by the expert so that you could do the solution of every question based on it.

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Before you study **midpoint theorem** let’s go through the syllabus of **CBSE class IX maths** which has the lessons, number system, the lessons in algebraic part are polynomial, linear equations in two variable, coordinate geometry part contains only one lesson, the lessons in geometry part are the introduction to Euclid’s geometry, lines, and angle, **triangles**, quadrilaterals, **area of quadrilateral and triangles,** circles and construction. Here one of the most important parts is **geometry**. In **CBSE IX class geometry** the** midpoint theorem** is taken from lesson number 3 ‘**Triangle’.** The **midpoint theorem** is one of the properties of the triangle, among all the theorem **midpoint theorem**, is the most important theorem. If you go through the last year’s question papers of maths you could see one or two questions based on **midpoint theorem** have been asked by the **CBSE board** every year. **Midpoint theorem** is based on the triangle which states that the line segment joining **midpoints** of two sides of the **triangle** is parallel to the third side and the line segment is half of the third side.

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**Midpoint theorem-**Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side.

**Given:**

ΔABC in which E and F are midpoints of AB and AC respectively.

#### To prove:

EF∥BC &

**Construction:**

Drawing CD∥AB and extending the line segment EF such that CD and EF intersect at D.

**Proof:**

CF = AF (Midpoint of AC)

∠DCF = ∠FAE (Alternative angles)

∠AEF = ∠FDC (Alternative angles)

ΔAEF ≅ ΔCDF (AAS rule)

∴ AE = DC ( By CPCT)

AE= BE (Midpoint of AB)

So,DC = BE

In the quadrilateral BCDE

DC = BE and DC∥ BE, If one set of opposite sides of a quadrilateral are equal and parallel then it is a parallelogram.

Therefore BCDE is a parallelogram

∴ BC∥ DE and thus

EF ∥ BC, Hence proved

Since we have proved above ΔAEF ≅ ΔCDF

EF = DF (By CPCT)

DE = BC (It has been proved above BCDE is a parallelogram)

EF + DF = DE

2EF = DE

2EF = BC

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**The converse of midpoint theorem:**

If a line is drawn through the midpoint of a side of the triangle which is parallel to another side then this line bisects the third side of the triangle.

In ΔPQR if T is the midpoint of PQ and TS∥ QR, then TS bisects PR

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