Mid Point Theorem

Introduction

The midpoint theorem in geometry helps in determining the sides of triangles that are missing values. It establishes a link between a triangle's sides and the line segment created by any two of the triangle's sides' midpoints. A line that connects two places has a midpoint in the middle of the line. The centre of a line is located halfway between its two endpoints. A line connecting these two places has a midpoint at the point where it is divided into two equal pieces. A second line that was drawn to divide the two regions in half also runs through the middle. In this tutorial, we will discuss the mid-point theorem.

Triangles

• Simple polygons with three sides and three interior angles make up triangles.

• One of the fundamental geometric shapes, it is represented by the symbol △ and consists of three connected vertices.

• Triangles can be categorised into a number of different varieties according to their sides and angles.

• Triangles are divided into the following types according to their side lengths.

Equilateral Triangle − When all three sides of a triangle are the same length, the triangle is said to be equilateral.

Isosceles triangle − An isosceles triangle is a triangle that has any two sides equal in length .

Scalene triangle − A scalene triangle is a triangle in which none of the sides are equal.

Mid Point Theorem Statement

According to the midpoint theorem, the line segment connecting any two triangle sides' midpoints are parallel to the third side and have a length equal to half of it.

Mid point theorem Proof by Congruency

Given that: D and E are mid points

To prove: $\mathrm{BC \parallel DE\:\: and\:\: DE = (\frac{1}{2} BC).}$

Prove:

Construction: Draw a line DF such that DE=EF

Compare triangle AED with triangle CEF −

• EC=AE (E is the midpoint of AC)

• $\mathrm{\angle FCE=\angle DAE }$(Alternate angles)

• $\mathrm{\angle FEC=\angle DEA }$(Vertically opposite angles)

By A.S.A. congruence we can say that both triangles are congruent

As a result, DE = EF and AD = CF. However, AD is equivalent to BD, and BD = CF and BD || CF. Hence, BCFD must be a parallelogram. Thus,

$$\mathrm{DF \parallel BC}$$

$$\mathrm{\Rightarrow DE \parallel BC}$$

$$\mathrm{and,DF = BC}$$

$$\mathrm{\Rightarrow DE + EF = BC }$$

$$\mathrm{\Rightarrow 2DE = BC \:\:\:\:\:\:\:\:\:\:\:\: (DE = EF\: proved)}$$

$$\mathrm{\Rightarrow DE =(\frac{1}{2} BC).}$$

Mid point theorem Proof by Similarity

In triangle ADE and triangle ABC

$$\mathrm{\angle BAC = \angle DAE}$$ Common angle

$$\mathrm{\frac{AB}{AD}=\frac{1}{2} \:\:\: (Given\: that\: D\: is\: mid\: point)}$$

$$\mathrm{\frac{AC}{AE}=\frac{1}{2}\:\:\:\: (Given\: that\: E\: is\: mid\: point)}$$

From SAS rule we can say that both triangles are similar since the similarity ratio is 2: 1we can say that the length of DE is half of the length of BC. Also, in similar triangles $\mathrm{\angle ABC =\angle ADE}$ (corresponding angles are equal)

$$\mathrm{\Rightarrow DE \parallel BC}$$

Important Notes

According to the midpoint theorem, the line segment connecting any two triangle sides' midpoints is parallel to the third side and equal to half of its length.

According to the reverse of the midpoint theorem, a line drawn through the middle of one triangle's side and parallel to another side will cut through the third side.

Applications of Mid point theorem

The mid-point theorem in geometry helps in identifying the missing values on a triangle's sides. It defines the relationship between the triangle's sides and the segment of the line drawn between any two locations in the triangle's middle. According to the midpoint theorem, a section of a line drawn through the centre of every triangle is parallel to the third side's breadth. According to the midpoint theorem, the segment of a line that connects any triangle's points is parallel to and equal to half of the third dimension. As an illustration, think about the ABC triangle. Let D and E represent the sites where AB and AC meet in the middle, respectively. Consider joining D and E. From a midway perspective, DE will be equal to half of BC and aligned with it.

Solved Examples

Example 1: Show that the given triangles are similar triangles.

Solution: In triangle ABC and XYZ

$$\mathrm{\angle B= \angle Z=90\: degrees}$$

$$\mathrm{\frac{AB}{XY}=\frac{BC}{YZ}=\frac{2}{3}}$$

From SAS rule we can say that both triangles are similar to each other.

Example 2: Find x if the Q and P are mid points and the length of PQ is 2x-4 and the length of BC is 32.

Solution: It is given that Q and P are mid points and the length of PQ is 2x-4 and the length of BC is 32 then according to midpoint theorem we can say that PQ and BC are parallel to each other and the length of PQ is half of the length of the BC that is $\mathrm{PQ=\frac{1}{2} BC }$

Now put their values $\mathrm{2x-4 =\frac{1}{2}×32 }$

$$\mathrm{\Rightarrow 2x-4 =16 }$$

$$\mathrm{\Rightarrow 2x=20 }$$

$$\mathrm{\Rightarrow x=10 }$$

Hence the value of x is 10 and the length of PQ is 16.

Example 3: Find the length of DE and AC if D and E are the mid points.

Solution: It is given that D and E are the mid points and the length of DE is 3x+6 and the length of AC is 12x-6 then according to mid point theorem we can say that DE and AC are parallel to each other and the length of DE is half of the length of the AC that is $\mathrm{DE=1/2\frac{1}{2}AC}$

Now put their values $\mathrm{3x+6=\frac{1}{2}×(12x-6) }$

$$\mathrm{\Rightarrow 3x+6=6x-3 }$$

$$\mathrm{\Rightarrow 3x=9 }$$

$$\mathrm{\Rightarrow x=3}$$

Hence the value of x is 3 and the length of DE is 15

and the length of AC is 30.

Example 4: What kind of triangle is it, if each of its angles is less than 90 degrees?

Solution: A triangle is referred to as an acute triangle if all of its angles are smaller than 90°.

Conclusion

Polygons with three sides and three interior angles make up triangles. According to the midpoint theorem, the line segment connecting any two triangle sides' midpoints are parallel to the third side and have a length equal to half of it.

FAQs

1. What do you mean by a triangle?

Polygons with three sides and three interior angles make up triangles.

2. What is the statement of the midpoint formula?

According to the midpoint theorem, the line segment connecting any two triangle sides' midpoints is parallel to the third side and equal to half of its length.

3. What is the converse of the midpoint formula?

According to the converse (inverse) of the midpoint theorem, "if a line is drawn across the midpoint of one side of a triangle, and parallel to the second side, it bisects the third side."

4. What do you mean by midpoint?

The midpoint is the point on a line segment that is closest to both endpoints and is also the centre.

5. What is the use of midpoint formula?

The mid-point theorem in geometry helps in identifying the missing values on a triangle's sides. It defines the relationship between the triangle's sides and the segment of the line drawn between any two locations in the triangle's middle.