Midpoint: Definition, Derivation, Formula, Properties & Solved Examples

Midpoint: Definition, Derivation, Formula, Properties & Solved Examples

A midpoint is defined as the point in the middle of the line joined by two points.

In a line segment, a point that bisects the line in two equal parts is the midpoint. If we draw a line between the line segment, dividing the line into two parts always passes through the midpoint.

In co-ordinate geometry, the midpoint formula is used to determine the line's center point whose coordinates are given to us.

The midpoint formula helps us to find the coordinates of the points if the other coordinates and midpoints are known to us.

Students whose basics are not cleared can prefer RD Sharma's solution for class 10.

What is Midpoint Formula A and B are the two end points in the coordinate axes having A ( 𝑥1𝑦1) B( 𝑥2𝑦2).

The mid point formula is defined as half of the sum of x-cordinates and half of the sum of the y-coordinates in point A and B. Refer to Important questions for class 10 maths to solve examples on midpoint formula.

Midpoint formula =

𝑥1 + 𝑥2 2

,

𝑦1 + 𝑦2 2

Midpoint Formula Derivation Consider two end points A & B of the line AB in coordinate axes whose endpoints are A ( 𝑥1𝑦1) B(𝑥2𝑦2).

The midpoint is defined as the average of the axes of both points.

1. For X-axis

For X-axis the midpoint is

𝑥1 + 𝑥2 2

.

2. For Y-axis

For Y-axis the midpoint is So the midpoint formula is

𝑦1 + 𝑦2 2

𝑥1 + 𝑥2 2

𝑦1 + 𝑦2

,

2

.

How to Find Midpoint There are two methods to find the midpoint of any two points in coordinate axes.

1. By using formula Find the midpoint of the line whose end points are (-4,5), (6,7). 𝑥1 = -4

𝑦1 = 5

𝑥2 = 6

𝑦2 = 7

midpoint formula =

𝑥1 + 𝑥2 2

,

𝑦1 + 𝑦2 2

=

−4 + 6 2

,

5+7 2

= (1, 6) The midpoint is (1, 6).

2. If the line segment of the given point is vertical or horizontal then divide the length by 2 and count the value from any of the end point gives you the mid point of the line.

Let’s take the same example to understand.

The coordinates of the points are (-4,5), (6,7) According to the method the length of the horizontal line 7+5 = 12.

Divide the length by 2 = 6 Moving 6 units from (-4,5) will give the midpoint 1,6 for the line segment.

Properties of Midpoint ● The midpoint divides the line segments into two equal(1:1) ratios. ● It divides the line segment into two equal parts. ● The bisector of the line segment is passed through the midpoint.

Use of Midpoint Formula To Find centroid of the Triangle The intersection point of all the three medians of the triangle is called the centroid of the triangle. It is the center point of the triangle.

The line segment from the vertex joining the midpoint of the opposite side of the triangle is called the median. The median divides the triangle in a ratio of 2:1.

Let a triangle whose vertices are 𝑥1𝑦1, 𝑥2𝑦2, 𝑥3𝑦3 AD, BE, CF are the median of the triangle F, E, D are the midpoints of the triangle ABC G is centroid of the triangle and it divides the triangle in 2:1 ratio.

So the centroid of the triangle is

𝑥1+𝑥2+𝑥3 3

,

𝑦1+𝑦2+𝑦3 3

Section Formula Section formula is used to find the coordinates of the points when it divides the line segments in some ratios.

When a point divides the line segments in different ratios then we use section formula.

Lets a point P(x,y) divides the line segments AB of points A( 𝑥1𝑦1) and B(𝑥2 𝑦2).

Methematical representation of the section formula is

𝑚𝑥2+𝑛𝑥1

P(x,y) =

𝑚+𝑛

,

𝑚𝑦2+𝑛𝑦1 𝑚+𝑛

Solved Examples on Midpoint Formula 1. Find the midpoint of the line whose endpoints are (6,4) and (5,6). 𝑥1 = 6

𝑦1 = 4

𝑥2 = 5

𝑦2 = 6

midpoint formula = =

6+5 2

,

𝑥1 + 𝑥2 2

,

𝑦1 + 𝑦2 2

4+6 2

= (5.5, 5) The midpoint is (5.5, 5).

2. If (2, 0) is the midpoint of the line joining the points A(6, 5) and B, find the coordinates of B. Given (2, 0) are the midpoint of A and B. A = (6,5) which is equals to 𝑥1𝑦1 Let the coordinates of B are(𝑥2𝑦2) midpoint formula =

𝑥1 + 𝑥2 2

,

𝑦1 + 𝑦2 2

(2,0) = 2=

6 + 𝑥2 2

,

5 + 𝑦2 2

6 + 𝑥2 2

4 = 6+ 𝑥2 𝑥2 = -2 now 0=

5 + 𝑦2 2

0 = 5 + 𝑦2 𝑦2 = -5 So the coordinates of y axes are(-2,-5).

Frequently Asked Questions(FAQs) 1. What is midpoint?

The midpoint is the point between the line segments which divides the line segments into two equal parts.

2. What is the midpoint theorem?

The line segment of the triangle which joins the two sides of the triangle is always parallel to the third side of the triangle.

3. What is the midpoint formula in coordinate geometry?

The midpoint formula in coordinate geometry is used to find the midpoint of the straight line by using the coordinates of both endpoints.

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