## An Introduction to Midpoint Theorem

Midpoint lies in the middle of a line connecting two points. A line has two endpoints, and the centre is in the middle of those two endpoints. A midpoint is a point at which a line connecting these two points is divided into two equal parts. In addition, a line drawn to double divide the line connecting the two areas passes through the centre.

## Midpoint Theorem

The Midpoint Theorem states that the line segment connecting the midpoints of any two sides of a triangle is parallel to the 3^{rd} side and equal to half of the 3^{rd} side. For example, let’s take an arbitrary triangle, ΔABC. Suppose that D and E are the midpoints of line segments AB and AC. Now, assuming that you join D and E:

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The statement of the Midpoint Theorem declares that DE will be parallel to BC and also equivalent to exactly half of BC.

### Midpoint Theorem Proof

To state and prove the midpoint theorem, we need to do the following:

According to the Mid-Point Theorem, a segment connecting the mid-points of two sides of a triangle is parallel to the 3^{rd} side and equal to half the 3^{rd} side.

Given: In ΔABC, P and Q are mid-points of line segments AB and AC, respectively.

To Prove:

i) PQ || BC

ii) PQ = ½ BC

Construction: Draw BA || CR to meet PQ generated at R.

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Proof:-

In ΔAEF and ΔCDF,

1. AF=CF (F is midpoint of side AC) ------ (1)

2. ∠AFE= ∠CFD (Vertically opposite angles) ------ (2)

3. ∠EAF= ∠DCF ------ (3) (Alternate angles, BA II CD (through construction), AC is a transversal) Alternate.angles,BASICS(through construction),ACisatransversal

4. Thus, ΔAEF = CDF (by rule of ASA) ------ (4)

5. EF = FD and AE = CD (by CPCT) ------ (5)

6. AE = BE (E is midpoint of side AB) ------ (6)

7. BE = CD (from equation 5 and 6) ------ (7)

8. EBCD is a parallelogram IIgm BAIICD (through construction) and BE=CD (from7) BA II CD(through construction) and BE=CD (from 7)

9. EF II BC and ED = BC (Because EBCD is a IIgm) ------ (8)

10. EF = ½ ED 1/2 ED (Because, EF = FD (from 5)

11. EF = ½ BC 1/2 BC (Because, ED = BC (from 8)

Therefore, EF II BC and EF = ½ BC prove the mid-point theorem.

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### What is the Converse of the Midpoint Theorem?

The converse of the centre theorem states that "passing the centre of 1 aspect of a triangle and drawing a line parallel to the opposite aspect bisects the third aspect".

### Uses of Midpoint Theorem

In geometry, the mid-point theorem helps us determine the missing values on the sides of a triangle. It establishes the relationship between the sides of the triangle and the part of the line drawn between any points in the middle of the triangle. According to the midpoint theorem, a section of a line drawn in the centre of any triangle is parallel to the length of the third side of the triangle.

According to the midpoint theorem, the part of the line that connects the points between any of the triangles is parallel to the third dimension and is equal to half of the third dimension. Consider the ABC triangle as an example. Let D and E be the intermediate points of AB and AC, respectively. Imagine connecting D and E. From a midpoint point of view, DE will be aligned with BC and equal to half BC.

A straight line connecting the points between any of the triangular sides is considered to be parallel and half the length of the third dimension.

The dialogue of the centre point theory assumes that a line drawn in the centre of one of the sides of a triangle and parallel to the other side will double divide the third side.

For basic calculations, we have a midpoint theorem formula, with applications in coordinate geometry, also known as the midpoint theorem of the line segment. It states that if we have a line segment with links (x_{1}, y_{1}) and (x_{2}, y_{2}) on both ends, we can find links in the middle and part of the line.

### Solved Examples on the Midpoint Theorem

Example 1: Take a ΔABC, and suppose that D is any point on the side BC. Assume that X and Y are the midpoints of sides AB and AC, respectively. Prove that XY will intersect AD.

Solution: By the statement of midpoint theorem, XY || BC. Now, consider triangle ABD. The line segment XE is parallel to the base of the side BD, and X is the midpoint of side AB. Through the converse of the midpoint theorem statement, E should be the midpoint of AD.

Hence, XY intersects AD.

Example 2: In the adjoining figure, all measurements are reflected in centimetres. Find the length of the side AO if AX measures 10 cm.

Solution: Given in the figure, AD = DB = 4cms

AE = EC = 6cms

Then, D and E are the midpoints of AB and AC, respectively.

Thus, DE intersects AX at point O.

Therefore, AO = ½ AX

= 1/2 × 10 = 5

Thus, the length of AO = 5.

## FAQs on Midpoint Theorem

**1. What do we understand by the converse of the Midpoint Theorem?**

In Euclidean geometry, the converse of the midpoint theorem is as crucial as the theorem itself. The converse of the midpoint theorem states that if a line segment is drawn crossing across the midpoint of any one side of a triangle and parallel to another side, then this line segment intersects the remaining 3^{rd} side.

**2. What is meant by the Midpoint Theorem?**

The midpoint is the centermost point of a line segment, which also means that it is at the same distance from both endpoints.

**3. What is the use of the Midpoint Theorem Formula?**

The midpoint formula is used to identify the coordinates of the midpoint of the line segment. This formula used for the midpoint theorem is familiar to high school students.

The point that is equidistant from two points, A (x_{1}, y_{1}) and B (x_{2}, y_{2}), on a line segment, is known as the midpoint.