Welcome

JAWAHAR NAVODAYA VIDYALAYA

LEPAKSHI

ANANTAPUR DISTRICT

COORDINATE GEOMETRY

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CLASS :X

CBSE

TOPIC: Section Formula

BY G.SUMATHI

TGT MATHEMATICS

JNV ANANTAPUR

ORDER OF SLIDES

• Section Formula
• Derivation of section formula
• Midpoint formula
• Examples
• Points to remember
• Summary
• Homework

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0 1 2 3 4 5 …

0 1 2 3 4 5 …

Section formula

A

B

(x ,y)

(x₁,y₁)

(x₂,y₂)

m ₁

m₂

Section formula

(x₁,y₁)

(x₂,y₂)

P

(x ,y)

A

B

m ₁

m₂

X =

Y =

Section formula

O

(x ,y)

(x₂,y₂)

(x₁,y₁)

m₁

m₂

_C

_Q

_R

_S

_T

_P

_B

_A

• Draw AR, PS and BT perpendicular to the

x-axis..

by AA similarity criterion,

• Draw AQ and PC parallel to the x-axis

• ΔPAQ ~ ΔBPC

⁼

⁼

⁼

⁼

x

x₂

x₁

y₁

y₂

y

SECTION FORMULA

• = =

​

• m₂(x-x₁) = m₁(x₂-x)
• m₂x-m₂x₁ = m₁x₂-m₁x
• m₁x₂-m₁x = m₂x-m₂x₁
• m₁x₂ + m₂x₁= m₂x +m₁x
• m₁x₂ + m₂x₁= (m₂ +m₁)x
• x =

​

​

​

​

Similarly

y =

A

P

B

(x ,y)

(x₁,y₁)

(x₂,y₂)

m₁

m₂

Section formula

SECTION FORMULA

• If point P(x, y) divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂), internally, in the ratio m₁:m₂ then coordinates of P(x, y) are

(x,y) =( , )

​

• Other form
• x = , y =
• If P(x ,y) is mid point then m₁:m₂= 1:1,the formula

​

x = y=

​

​

​

Section formula

A

P

B

(x ,y)

(x₁,y₁)

(x₂,y₂)

m₁

m₁

Mid point formula

EXAMPLE1:

Q) Find the co-ordinates of the point which divides line segment joining (-1, 7) & (4,-3)in the ratio 2:3.

Sol: Given x₁ = -1, x₂=4 ,y₁= 7 y₂= -3 ,m₁= 2 & m₂= 3

section formula

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x= 1 y= 3

​

​

x=

y =

x=

y =

Hence the coordinates of the required point = (1,3)

x=

y =

EXAMPLE2:

Q) Find the ratio in which the line segment joining the points (-3, 10) & (6,-8) is divided by (-1, 6).

Sol: Given x₁ = -3, x₂=6 ,y₁= 10 y₂= -8 ,x=-1 & y= 6

section formula

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-m₁-m₂ = 6m₁ - 3m₂

- 6m₁- m₁ = -3m₂ + m₂

-7m₁ = -2m₂

​

​

A(-3,10)

P (-1,6)

B(6,-8)

​

m₁

m₂

(x₁,y₁)

(x₂,y₂)

(x, y)

x=

y =

-1=

EXAMPLE3:

Q) If A(1, 2),B(4,y),C(x,6) and D(3,5) are the vertices of a parallelogram taken in order, find x and y.

Sol:

​

​

• In a parallelogram diagonal bisect each other
• Mid points AC = Mid points BD
• Mid point formula = [ ]

​

A(1,2)

B(4,y)

C(x,6)

D(3,5)

O

EXAMPLE3 CONTD…

Sol:

• Mid points AC = [ , ] = [ , 4 ]

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• Mid points BD = [ ] = [ , ]
• Mid points AC = Mid points BD
• [ ,4] = [ , ]

​

• = and 4 =
• x = 6 and y= 3

A(1,2)

B(4,y)

C(x,6)

D(3,5)

O

POINT TO REMEMBER:

• Coordinates of origin are (0,0)
• Coordinates of point on x-axis are (x,0)
• Coordinates of point on y-axis are (0,y)
• Points of trisection:

Two points dividing the line segment into three equal parts

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For finding P , m₁:m₂ = 1:2

For finding Q , m₁:m₂ = 2:1

​

​

A

Q

B

P

SUMMERY

• If point P(x, y) divides the line segment joining the points A(x₁, y₁) and B(x₂, y₂), internally, then coordinates of P(x, y) are

(x,y) =( , )

​

• Other form
• x = , y =
• If P(x ,y) is mid point then m₁:m₂= 1:1,the formula

​

x = y=

​

​

​

Section formula

A

P

B

(x ,y)

(x₁,y₁)

(x₂,y₂)

m₁

m₁

Mid point formula

HOME WORK:

• 1.Prove that the points (-2, -1), (1,0), (4,3) and (1,2 ) are the vertices of a parallelogram . Is it a rectangle?
• 2. Find the co-ordinates of the point of trisection of line segment joining (4,-1) and(-2,-3)?
• 3.Find the ratio in which the line segment joining A (1,-5) and B (-4, 5) is divided by the X-axis also find the coordinates of the point of division?
• Exercise 7.2 of NCERT Text Book

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THANK

YOU

ALL…

BY G.SUMATHI

TGT MATHEMATICS

JNV, ANANTAPUR