Section Formula is a useful tool in coordinate geometry, which helps us find the coordinates of any point on a line by dividing the line into some known ratio. In this article, we will learn about section formulas, the types of division of lines, and how to solve problems based on them in detail.

What is Section Formula?

Section Formula is a mathematical tool used in coordinate geometry to determine the coordinates of a point that divides a line segment joining two given points in a given ratio. It is particularly useful in finding points that partition a segment either internally or externally in a specified ratio.

There are two cases for section formula i.e.,

• Internal Section Formula
• External Section Formula

Internal Section Formula

Internal Section Formula

For a line segment joining A(x₁, y₁) & B(x₂, y₂) and we want to find the coordinates of a point P that divides the line segment AB internally in the ratio m:n. Then the coordinates of P are:

P(x, y) = {(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)}

External Section Formula

External Section Formula

For a line segment joining A(x₁, y₁) & B(x₂, y₂) and we want to find the coordinates of a point P that divides the line segment AB externally in the ratio m:n. Then the coordinates of P are:

P(x, y) = {(mx_{2 –} nx₁)/(m – n), (my_{2 –} ny₁)/(m – n)}

Midpoint Formula

For a line AB with coordinates A(x₁, y₁) and B(x₂, y₂), then coordinates of midpoint is given as

M(x, y) ={(x₁ + x₂)/2, (y₁ + y₂)/2}

Note: Midpoint formula is the special case section formula where midpoint divides the line in 1:1.

Section Formula Practice Questions with Solution

These are some important Section Formula Practice Questions with Solution

Question 1: The point P divides the line segment AB joining points A(2, 1) and B(-3, 6) in the ratio 2:3.

Solution:

Given that A(2, 1)=(x₁, y₁), B(-3, 6) = (x₂, y₂)

Point P divides the segment AB in the ratio 2:3, hence m = 2, n = 3

Formula: P(x, y) = {(mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)}

Substituting all the known values,

P(x, y) ={[(2(-3)+3(2))/(2+3)],[(2(6)+3(1))/(2+3)]}

⇒ P(x, y) =[(-6+6/5), (12+3/5)]

⇒ P(x, y) = (0/5, 15/5)

⇒ P(x, y) = (0, 3)

Question 2: A (4, 5) and B(7, – 1) are two given points and the point Y divides the line-segment AB externally in the ratio 4:3. Find the coordinates of Y.

Solution:

Given that, A(4, 5) = (x₁, y₁), B(7, -1) = (x₂, y₂)

Point Y divides the segment AB in the ratio 4:3, hence m = 4, n = 3

Formula: Y(x, y) = {(mx_{2 –} nx₁)/(m – n), (my_{2 –} ny₁)/(m – n)}

Substituting the known values,

Y(x, y) = {[(4(7)-3(4))/(4-3)],[(4(-1)-3(5)/(4-3)]}

⇒ Y(x, y) = {(28-12)/1,(-4-15)/1} ={16,-19}

The coordinates for the point Y are (16,-19).

Question 3: Find the midpoint of AB where A(3,4) and B(5,7).

Answer:

Given that, A(3, 5) = (x₁, y₁), B(5, 7) = (x₂, y₂)

Formula: M = {(x₁ + x₂)/2, (y₁ + y₂)/2}

Substituting the known values,

M = {(3 + 5)/2,(5 + 7)/2} ={8/2, 12/2} =(4, 6)

The Midpoint of the AB is (4, 6).

Question 4: Find the midpoint of AB where A(1, 4) and B(5, 8).

Answer:

Given that, A(1, 4)=(x₁, y₁), B(5, 8) = (x₂, y₂)

Formula: P = {(x₁ + x₂)/2,(y₁ + y₂)/2}

Substituting the known values,

P={(1 + 5)/2,(4 + 8)/2} = {6/2, 12/2} = (3, 6)

The Midpoint of the AB is (3,6).

Question 5: If a point P(k, 7) divides the line segment joining A(8, 9) and B(1, 2) in a ratio m : n then find values of m and n.

Solution:

Given coordinates are A (8, 9) and B (1, 2)

Let the given point P (k, 7) divides the line segment in the ratio of m : 1

Using section formula for y coordinate.

⇒ 7 = (my₂ + ny₁)/(m + n )

⇒ 7 = (m × 2 + 1 × 9)/(m + 1)

⇒ 7 = (2m + 9)/(m +1)

⇒ 7m + 7 = 2m +9

⇒ 5m = 2

⇒ m = 5 / 2

So the required ratio is 5 : 2

Therefore, value of m is 5 and value of n is 2.

Question 6: If a point P(k, 2) divides the line segment joining A(6, 8) and B(2, 3) in a ratio m : n then find values of m and n.

Solution:

Given coordinates are A (6, 7) and B (2, 2)

Let the given point P (k, 2) divides the line segment in the ratio of m : 1

Using section formula for y coordinate.

⇒ 2 = (my₂ + ny₁)/(m + n )

⇒ 7 = (m × 2 + 1 × 8)/(m + 1)

⇒ 7 = (2m + 8)/(m +1)

⇒ 7m + 7 = 2m +8

⇒ 5m = 1

⇒ m = 1/5

So the required ratio is 1 : 5

Therefore, value of m is 1 and value of n is 5.

Section Formula practice Questions: Unsolved

Question 1: The point P divides the line segment AB joining points A(-2, 1) and B(-3, 6) in the ratio 2:3.

Question 2: A (5, 6) and B(2, – 1) are two given points and the point Y divides the line-segment AB externally in the ratio 5:3. Find the coordinates of Y.

Question 3: The point P divides the line segment AB joining points A(5, 1) and B(-3, 6) in the ratio 1:1.

Question 4: If a point P(2, p) divides the line segment joining A(8, 5) and B(2, 3) in a ratio m : n then find values of m and n.

Question 5: The point P divides the line segment AB joining points A(-2, -1) and B(-3, -9) in the ratio 2:1.

Question 6: If a point P(k, 3) divides the line segment joining A(4, 8) and B(5, 3) in a ratio m : n then find values of m and n.

Question 7: A (-4, 5) and B(7, 1) are two given points and the point Y divides the line-segment AB externally in the ratio 2:3. Find the coordinates of Y.

Question 8: If a point P(k, 4) divides the line segment joining A(2, 9) and B(1, 3) in a ratio m : n then find values of m and n.

Read More,

• Section Formula
• Midpoint Formula
• Distance Formula
• Euclidean Distance

Section Formula practice Questions – FAQs

What is  Section Formula?

Section Formula is a tool in coordinate geometry used to find the coordinates of a point that divides a line segment in a specific ratio, either internally or externally.

Write Internal Section Formula.

If a point C divides a line joining A(x₁, y₁) and B(x₂, y₂) internally in a ratio of m:n, then section formula is given as follows:

[Tex]C(x,y)= (mx_2+nx_1)/m+n, (my_2+ny_1)/m+n[/Tex]

What is External Division of line?

When a point lies on the extended line segment to divide it in some ratio externally, it is called external division of line.

What is the special case of the Section Formula for the Midpoint?

The case of midpoint of line segment is the special case of the section formula, and coordinates of midpoint of a line segment joining A(x₁, y₁) and B(x₂, y₂), is given as follows:

[Tex]C(x,y)=(x_1+x_2)/2,(y_1+y_2)/2[/Tex]