This document provides an introduction to coordinate geometry. It defines the Cartesian coordinate system using perpendicular x and y axes with points located by their coordinates (x,y). It describes the four quadrants and uses the distance formula to calculate distances between points. It also discusses applications of the distance formula like finding midpoints and using it to classify triangles and quadrilaterals. The section formula is introduced to find the coordinates of a point dividing a line segment internally in a given ratio. Lastly, the area of a triangle is defined using trapeziums formed by the perpendiculars from the vertices to the x-axis.

1. Coordinate Geometry
Class X

2. 1. Cartesian Coordinate
system and Quadrants
2. Distance formula
3. Area of a triangle
4. Section formula
Objectives

3. • The use of algebra to study
geometric properties i.e.
operates on symbols is defined
as the coordinate system.
What is co-ordinate geometry ?
IntroductIon

4.  A system of geometry where the position of points on the plane is described
using an ordered pair of numbers.
 The method of describing the location of points in this way was
 proposed by the French mathematician René Descartes .
 He proposed further that curves and lines could be described by
equations using this technique, thus being the first to link
algebra and geometry.
 In honor of his work, the coordinates of a point are often referred to as
its Cartesian coordinates, and the coordinate plane as the Cartesian
Coordinate Plane.
Coordinate Geometry

5. SoME BASIc PoIntS
To locate the position of a point on a plane,we
require a pair of coordinate axes.
The distance of a point from the y-axis is called its
x-coordinate,OR abscissa.
The distance of a point from the x-axis is called its
y-coordinate,OR ordinate.
The coordinates of a point on the x-axis are
of the form (x, 0) and of a point on the y-axis
are of the form (0, y).

6. RECAP
Coordinate Plane
X
X’
Y
Y’
O
Origin
1 2 3 4
+ve direction
-1
-2
-3
-4
-ve direction
-1
-2
-3
-ve
direction
1
2
3
+ve
direction
X-axis : X’OX
Y-axis : Y’OY

7. Quadrants
X
X’ O
Y
Y’
I
II
III IV
(+,+)
(-,+)
(-,-) (+,-)

8. Coordinates
X
X’
Y
Y’
O
1 2 3 4
-1
-2
-3
-4
-1
-2
-3
1
2
3
(2,1)
(-3,-2)
Ordinate
Abcissa
(?,?)

9. X
X’ O
Y
Y’
I
II
III IV
(+,+)
(-,+)
(-,-) (+,-)
Q : (1,0) lies in which Quadrant?
Ist? IInd?
A : None. Points which lie on the axes do not lie in any
quadrant.

10. Distance Formula
0
x1
x2
y2
y1
P(x1,y1)
Q(x2,y2)
Y-axis
X-axis

11.  Let us now find the distance between any two points P(x1, y1) and
Q(x1, y2)
 Draw PR and QS ⊥ x-axis. A
perpendicular from the point P on
 QS is draw to meet it at the point T
So, OR = x1 , OS = x2 ,
PR = PS = y1 , QS = y2
Then , PT = x2 – x1 ,
QT = y2 – y1
x
Y
P (x1 , y1)
Q(x2 , y2)
T
R S
O
Distance Formula

12. Now, applying the Pythagoras theorem in ΔPTQ, we get
Therefore
2
2
2
QT
PT
PQ +
=
( ) ( )2
1
2
2
1
2 y
y
x
x
PQ −
+
−
=
which is called the distance formula.

13. Example 1: Find the distance between P(1,-3) and Q(5,7).
The exact distance between A(1, -3) and B(5, 7) is

14. Distance From Origin
Distance of P(x, y) from the origin is
( ) ( )
2 2
x 0 y 0
= − + −
2 2
x y
= +

15. Applications of Distance Formula
To check which type of triangle is
formed by given 3 coordinates.
and
To check which type of quadrilateral is
formed by given 4 coordinates.

16. Applications of Distance Formula
Parallelogram
Prove opposite sides are equal or diagonals bisect
each other

17. Applications of Distance Formula
Rhombus
Prove all 4 sides are equal

18. Applications of Distance Formula
Rectangle
Prove opposite sides are equal and diagonals are
equal.

19. Applications of Distance Formula
Square
Prove all 4 sides are equal and diagonals are equal.

20. Collinearity of Three Points
Use Distance Formula
a b
c
Show that a+b = c

21. Section Formula
2
1
m
m
PB
PA
=
 Consider any two points A(x1 , y1) and B(x1 ,y2) and assume that P (x, y)
divides AB internally in the ratio m1: m2 i.e.
 Draw AR, PS and BT ⊥ x-axis.
 Draw AQ and PC parallel to the x-axis.
Then,
by the AA similarity criterion,
x
Y
A (x1 , y1)
B(x2 , y2)
P (x , y)
R S
O T
m1
m2
Q
C

22. Section Formula
ΔPAQ ~ ΔBPC
---------------- (1)
Now,
AQ = RS = OS – OR = x– x1
PC = ST = OT – OS = x2– x
PQ = PS – QS = PS – AR = y– y1
BC = BT– CT = BT – PS = y2– y
Substituting these values in (1), we get
BC
PQ
PC
AQ
BP
PA
=
=
( )
( )
( )
( )
y
y
y
y
x
x
x
x
m
m
−
−
=
−
−
=
2
1
2
1
2
1 ( )
( )
( )
( )
y
y
y
y
x
x
x
x
m
m
−
−
=
−
−
=
2
1
2
1
2
1

23. Section Formula
For x - coordinate
Taking
or
( )
( )
x
x
x
x
m
m
−
−
=
2
1
2
1
( ) ( )
1
2
2
1 x
x
m
x
x
m −
=
−
1
2
2
1
2
1 x
m
x
m
x
m
x
m −
=
−
or
or ( )
1
2
1
2
2
1 m
m
x
x
m
x
m +
=
+
1
2
1
2
2
1
m
m
x
m
x
m
x
+
+
=

24. Section Formula
For y – coordinate
Taking ( )
( )
y
y
y
y
m
m
−
−
=
2
1
2
1
( ) ( )
1
2
2
1 y
y
m
y
y
m −
=
−
1
2
2
1
2
1 y
m
y
m
y
m
y
m −
=
−
( )
1
2
1
2
2
1 m
m
y
y
m
y
m +
=
+
1
2
1
2
2
1
m
m
y
m
y
m
y
+
+
=
or
or
or

25. Midpoint
Midpoint of A(x1, y1) and B(x2,y2)
m:n ≡ 1:1
1 2 1 2
x x y y
P ,
2 2
+ +
 
∴ ≡  
 
Find the Mid-Point of P(1,-3) and Q(5,7).

26. Area of a Triangle
 Let ABC be any triangle whose vertices are A(x1, y1), B(x2 , y2) and
C(x3 , y3).
 Draw AP, BQ and CR
 perpendiculars from A,B and C,
 respectively, to the x-axis.
Clearly ABQP, APRC and
BQRC are all trapezium,
Now, from figure
QP = (x2 – x1)
PR = (x3 – x1)
QR = (x3 – x2)
x
Y
A (x1 , y1)
B(x2 , y2)
C (x3 , y3)
P Q
O R

27. Area of a Triangle
X
X’
Y’
O
Y A(x1, y1)
C(x3, y3)
B(x
2
,
y
2
)
M L N
Area of ∆ ABC =
Area of trapezium ABML + Area of trapezium ALNC
- Area of trapezium BMNC

28. Area of a Triangle
Area of ABC
Δ = Area of trapezium ABQP + Area of
trapezium BQRC– Area of trapezium APRC.
We also know that ,
Area of trapezium =
Therefore,
Area of ABC
Δ =
( )( )
em
between th
distance
sides
parallel
of
sum
2
1
( ) ( ) ( )PR
CR
+
AP
2
1
CR
+
BQ
2
1
QP
AP
+
BQ
2
1
−
+ QR
( )( ) ( )( ) ( )( )
1
3
3
1
2
3
3
2
1
2
1
2
2
1
2
1
2
1
x
x
y
y
x
x
y
y
x
x
y
y −
+
−
−
+
+
−
+
=
( ) ( ) ( )
[ ]
1
3
3
3
1
1
3
1
2
3
3
3
2
2
3
2
1
1
2
1
1
2
2
2
2
1
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y
x
y −
+
−
−
−
+
−
+
−
+
−
=
( ) ( ) ( )
[ ]
1
2
3
3
1
2
2
3
1
2
1
y
y
x
y
y
x
y
y
x −
+
−
+
−
=
Area of Δ ABC

29. PROJECT
Mark coordinate axes on your city map and
find distances between important landmarks-
bus stand, railway station, airport, hospital, school,
Your house, Any river etc.

31. THANK YOU