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How much is the distance between
flower and the butterfly ?
x- axis
-5
-4

-3

-2

-1

0

1

2

3

4

5

6
2
3
Y- axis
1
0
-1

-1

-2

-2

-3

-3

Origin
0
1
(0,0)

-4

x- axis
-5
-4

2

3

4

5

6
3

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

1st quadrant

1

2

3

4

5

6
3

1st quadrant

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

2nd quadrant

1

2

3

4

5

6
3

1st quadrant

-3

-2

-1

0

1

2

3

4

5

6

-3

3rd quadrant

-2

-1

-4

-4

-5

0

1

2

2nd quadrant

4th quadran...
3
1

2

To mark a point on a plane (3, 0)

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

(3, 0)
1

2

3

4

5

6
3
2
1

To mark a point on a plane (-5, 0)

0

0

-1

-1

-2

-2

-3

-3

-4

(-5, 0)
-5
-4

1

2

3

4

5

6
3

-4

-3

-2

-1

0

1

-1

-5

0

1

2

( 0,3)

-4

-3

-2

To mark a point on a plane ( 0,3)

2

3

4

5

6
3

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

To mark a point on a plane ( 0,-1)

1
( 0,-1)

2

3

4

5

6
3
2
1

A

What are the co-ordinates of A ?
0

0

-1

-1

-2

-2

-3

-3

-4

-4

-5

1

2

3

4

5

6
3

To mark a point on a plane (4, 2 )

-4

-3

-2

-1

0

-1

-5

0

1

2

(4, 2)

1

2

3

-4

-3

-2

4 is x coordinate ...
2

3

(3,3)

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

To mark a point on a plane (3,3 )

1

2

3

4

5

6
3

(3,3)

1

2

1st quadrant

0

0

-1

-1

-2

-2

-3

-3

-4

-4

-5

1

2

3

4

5

6
3
1

2

(-3,3)

0

To mark a point on a plane (-3,3 )
-1

0
-1

-2

-2

-3

-3

-4

-4

-5

1

2

3

4

5

6
3

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

2nd quadrant (-3,3)

1

2

3

4

5

6
-2

-1

0

-1

-3

-2

(-4,-2)

-3

-4

-4

-5

0

1

2

3

To mark a point on a plane (-4,-2)

1

2

3

4

5

6
3
2
1

-2

-1

0

-3

0

-1

-4

-3

-2

(-4,-2)
3rd quadrant

-4

-5

1

2

3

4

5

6
-1

0

1

2

3

4

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

3

To mark a point on a plane (3,-2)

(3,-2)

5

6
3
2
1
0

-1

0

1

2

3

4

5

6

-1

-2

-2

-3

-3

-4

-4

-5

4th quadrant
(3,-2)
3

(-3,-1)

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

What is the coordinates of shown point?

1

2

3

4

5

6
3

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

What is the coordinates of shown point?

1

2

3

4

5

6
3

-1

0

-1

-2

-2

-3

-3

-4

-4

-5

0

1

2

What is the coordinates of shown point?

1

2

3

4

5

6
2

3

A

1

-3

-2

-1

0

-4

0

1

2

3

4

5

6

-3

-2

-1

What are the coordinates of vertices of the tria

-4

-5

...
3
2
1

-2

-1

0

-3

0

1

2

3

4

5

6

-1

-4

-3

-2

What are the coordinates of vertices of the star

-4

-5
3

(3,3)

2

To mark a point on a plane (3,3 )

-1

0 (0,0)1

-1

-2

-2

-3

-3

-4

-4

-5

0

1

Let there is a flower ...
(3,3)

Let there is a butterfly at (1,1)
(1,1)
(0,0)
How much is the distance between
flower and the butterfly ?
(1,1)

(3,3)
B
(3,3)

A (1,1)
O
Let A is the position of the Butterfly.A has co-ordinates (1,1).
B is the position of the flower.B has ...
B
(3,3)

C

A (1,1)
O

P

Q

Let us draw the perpendiculars AP on X axis ,BQ on Y axis.
AC on BQ to complete a right trian...
In Triangle ABC ,
the length of AC = PQ
PQ = OQ – OP =3 – 1 =2
the length of BC =BQ - CQ
=3 – 1 =2
(1,1)
A
In Right Triang...
B (x , y )
2
2

A

( x1 , y1 )

O

How much is the distance between
the points A and B ?
B
( x2 , y 2 )

( x1 , y1 )

A

O

P

C
Q

Perpendiculars AP and BQ on X axis are drawn..
AC on BQ are drawn to complete a...
B
( x2 , y 2 )

OP = x1

OQ = x2

y2 − y1

PQ = x2 - x1
AC = x2 - x1
BQ = y2
OQ = y1

BC = y2 - y1

y2

A
(x , y )
1

O

1...
B
( x2 , y 2 )

.

AC = x2 − x1
BC= y2 − y1

y2 − y1

A

( x1 , y1 )

O

x1 P

x2 − x1

x2

x2 − x1

C
Q
B
( x2 , y 2 )

AC = x2 − x1
BC= y2 − y1

AC = AB + BC
2

2

2

AC = AB2 + BC 2
AC = (x 2 − x1 ) 2 + (y 2 − y1 ) 2
d = (x ...
B (x , y )
2
2

It is called distance formula.

A

( x1 , y1 )

O

The distance between the points A ( x1 , y1 )
and B ( x...
B (x , y )
2
2

C is a point on the line joining
A and B in ratio

m:n
A

( x1 , y1 )

O
m:n

C ( x, y )

m

A and B in ratio

n

C is a point on the line joining

B (x , y )
2
2

A

( x1 , y1 )

What will be co...
n

( x2 , y2 )B
y2 − y

x2 − x

S

m

( x, y ) C
( x1 , y1 ) A

O

x1

x − x1

Q

R

N

M

x

y − y1

P

x2

Perpendicular...
n

( x2 , y2 )B

m

Triangle ACQ and
BCS are similar.
( x1 , y1 ) A

AC AQ
=
BC CS
O

x1

y2 − y

x2 − x

S

( x, y ) C

x...
n

( x2 , y2 )B

m

Triangle ACQ and
BCS are similar.
( x1 , y1 ) A

AC AQ
=
BC CS
O

x1

y2 − y

x2 − x

S

( x, y ) C

x...
This is called section formula B
C ( x, y )

m:n
A

m

A and B in ratio

n

C is a point on the line joining

( x2 , y 2 )...
in ratio

n

C is an exterior point point on
the line joining A and B

B (x , y )
2
2

m:n
A

C ( x, y )

m

( x1 , y1 )

...
Co-ordinates of circumcentre OF A TRIANGLE
WHEN VERTICES ARE GIVEN
B( x2 , y2 )

0
A
( x1 , y1 )

C

x1 + x2 + x3
x=
3

( ...
Co-ordinates of in -centre OF A
TRIANGLE WHEN VERTICES ARE GIVEN
B( x2 , y2 )

c

a

0
A
( x1 , y1 )

b

C

( x2 , y 2 )

...
ASSIGNMENT
Q1.Which point on x axis is equidistant from
(5,9) and (-4,6)?
Q2. Which point on y axis is
equidistant from (2...
ASSIGNMENT
Q3.Prove that (2a,4a),(2a,6a)and (2a+√3a) are vertices of an
equilateral triangle.
Q4.In what ratio the x-axis ...
ASSIGNMENT
Q5. For what value of x will the points (x,1),(2,1) and (4,5)
lie on a line?

Q6. Determine the ratio in which ...
ASSIGNMENT
Q7If the points (-2,-1), (1,0),(x,3) and ( 1,y) form a
parallelogram, find the value of x and y.
Q8.Find the co...
Thank You
Developed by Pratima Nayak,
Kendriya Vidyalaya,Fort Willia,Kolkata
Introduction to coordinate geometry by pratima nayak
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Introduction to coordinate geometry by pratima nayak

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The slide show helps me to introduce Coordinate Geometry in Secondary School level.

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Introduction to coordinate geometry by pratima nayak

  1. 1. How much is the distance between flower and the butterfly ?
  2. 2. x- axis -5 -4 -3 -2 -1 0 1 2 3 4 5 6
  3. 3. 2 3 Y- axis 1 0 -1 -1 -2 -2 -3 -3 Origin 0 1 (0,0) -4 x- axis -5 -4 2 3 4 5 6
  4. 4. 3 -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 1st quadrant 1 2 3 4 5 6
  5. 5. 3 1st quadrant -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 2nd quadrant 1 2 3 4 5 6
  6. 6. 3 1st quadrant -3 -2 -1 0 1 2 3 4 5 6 -3 3rd quadrant -2 -1 -4 -4 -5 0 1 2 2nd quadrant 4th quadrant
  7. 7. 3 1 2 To mark a point on a plane (3, 0) -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 (3, 0) 1 2 3 4 5 6
  8. 8. 3 2 1 To mark a point on a plane (-5, 0) 0 0 -1 -1 -2 -2 -3 -3 -4 (-5, 0) -5 -4 1 2 3 4 5 6
  9. 9. 3 -4 -3 -2 -1 0 1 -1 -5 0 1 2 ( 0,3) -4 -3 -2 To mark a point on a plane ( 0,3) 2 3 4 5 6
  10. 10. 3 -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 To mark a point on a plane ( 0,-1) 1 ( 0,-1) 2 3 4 5 6
  11. 11. 3 2 1 A What are the co-ordinates of A ? 0 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 1 2 3 4 5 6
  12. 12. 3 To mark a point on a plane (4, 2 ) -4 -3 -2 -1 0 -1 -5 0 1 2 (4, 2) 1 2 3 -4 -3 -2 4 is x coordinate or abscissa 2 is y coordinate or ordinate. 4 5 6
  13. 13. 2 3 (3,3) -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 To mark a point on a plane (3,3 ) 1 2 3 4 5 6
  14. 14. 3 (3,3) 1 2 1st quadrant 0 0 -1 -1 -2 -2 -3 -3 -4 -4 -5 1 2 3 4 5 6
  15. 15. 3 1 2 (-3,3) 0 To mark a point on a plane (-3,3 ) -1 0 -1 -2 -2 -3 -3 -4 -4 -5 1 2 3 4 5 6
  16. 16. 3 -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 2nd quadrant (-3,3) 1 2 3 4 5 6
  17. 17. -2 -1 0 -1 -3 -2 (-4,-2) -3 -4 -4 -5 0 1 2 3 To mark a point on a plane (-4,-2) 1 2 3 4 5 6
  18. 18. 3 2 1 -2 -1 0 -3 0 -1 -4 -3 -2 (-4,-2) 3rd quadrant -4 -5 1 2 3 4 5 6
  19. 19. -1 0 1 2 3 4 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 3 To mark a point on a plane (3,-2) (3,-2) 5 6
  20. 20. 3 2 1 0 -1 0 1 2 3 4 5 6 -1 -2 -2 -3 -3 -4 -4 -5 4th quadrant (3,-2)
  21. 21. 3 (-3,-1) -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 What is the coordinates of shown point? 1 2 3 4 5 6
  22. 22. 3 -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 What is the coordinates of shown point? 1 2 3 4 5 6
  23. 23. 3 -1 0 -1 -2 -2 -3 -3 -4 -4 -5 0 1 2 What is the coordinates of shown point? 1 2 3 4 5 6
  24. 24. 2 3 A 1 -3 -2 -1 0 -4 0 1 2 3 4 5 6 -3 -2 -1 What are the coordinates of vertices of the tria -4 -5 C B
  25. 25. 3 2 1 -2 -1 0 -3 0 1 2 3 4 5 6 -1 -4 -3 -2 What are the coordinates of vertices of the star -4 -5
  26. 26. 3 (3,3) 2 To mark a point on a plane (3,3 ) -1 0 (0,0)1 -1 -2 -2 -3 -3 -4 -4 -5 0 1 Let there is a flower at this point. 2 3 4 5 6
  27. 27. (3,3) Let there is a butterfly at (1,1) (1,1) (0,0)
  28. 28. How much is the distance between flower and the butterfly ? (1,1) (3,3)
  29. 29. B (3,3) A (1,1) O Let A is the position of the Butterfly.A has co-ordinates (1,1). B is the position of the flower.B has the coordinate ( 3, 3).
  30. 30. B (3,3) C A (1,1) O P Q Let us draw the perpendiculars AP on X axis ,BQ on Y axis. AC on BQ to complete a right triangle ABC.
  31. 31. In Triangle ABC , the length of AC = PQ PQ = OQ – OP =3 – 1 =2 the length of BC =BQ - CQ =3 – 1 =2 (1,1) A In Right Triangle ABC , O P 2 2 2 AC = AB + BC AC = AB 2 + BC 2 AC = 2 + 2 = 4 + 4 = 8 = 2 2 2 2 B (3,3) C Q
  32. 32. B (x , y ) 2 2 A ( x1 , y1 ) O How much is the distance between the points A and B ?
  33. 33. B ( x2 , y 2 ) ( x1 , y1 ) A O P C Q Perpendiculars AP and BQ on X axis are drawn.. AC on BQ are drawn to complete a right triangle ABC.
  34. 34. B ( x2 , y 2 ) OP = x1 OQ = x2 y2 − y1 PQ = x2 - x1 AC = x2 - x1 BQ = y2 OQ = y1 BC = y2 - y1 y2 A (x , y ) 1 O 1 x1 P x2 − x1 x2 x2 − x1 C y1 Q
  35. 35. B ( x2 , y 2 ) . AC = x2 − x1 BC= y2 − y1 y2 − y1 A ( x1 , y1 ) O x1 P x2 − x1 x2 x2 − x1 C Q
  36. 36. B ( x2 , y 2 ) AC = x2 − x1 BC= y2 − y1 AC = AB + BC 2 2 2 AC = AB2 + BC 2 AC = (x 2 − x1 ) 2 + (y 2 − y1 ) 2 d = (x 2 − x1 ) 2 + (y 2 − y1 ) 2 y2 − y1 ( x1 , y1 ) A O P x1 x2 − x1 C x2 − x1 Q x2
  37. 37. B (x , y ) 2 2 It is called distance formula. A ( x1 , y1 ) O The distance between the points A ( x1 , y1 ) and B ( x2 , y2 ) d = ( x2 − x1 ) + ( y2 − y1 ) 2 2
  38. 38. B (x , y ) 2 2 C is a point on the line joining A and B in ratio m:n A ( x1 , y1 ) O
  39. 39. m:n C ( x, y ) m A and B in ratio n C is a point on the line joining B (x , y ) 2 2 A ( x1 , y1 ) What will be co-ordinates of C?
  40. 40. n ( x2 , y2 )B y2 − y x2 − x S m ( x, y ) C ( x1 , y1 ) A O x1 x − x1 Q R N M x y − y1 P x2 Perpendiculars AM,CN , BP AR CS Are drawn. y2 − y1
  41. 41. n ( x2 , y2 )B m Triangle ACQ and BCS are similar. ( x1 , y1 ) A AC AQ = BC CS O x1 y2 − y x2 − x S ( x, y ) C x − x1 x2 Q y2 − y1 R N M x y − y1 P m x − x1 ⇒ = ⇒ mx2 − mx = nx − nx1 n x2 − x ⇒ mx + nx = nx1 + mx2 ⇒ x(m + n) = nx1 + mx2 nx1 + mx2 ⇒x= ( m + n)
  42. 42. n ( x2 , y2 )B m Triangle ACQ and BCS are similar. ( x1 , y1 ) A AC AQ = BC CS O x1 y2 − y x2 − x S ( x, y ) C x − x1 Q y2 − y1 R N M x y − y1 P x2 m y − y1 ⇒ = ⇒ my2 − my = ny − ny1 n y2 − y ⇒ my + ny = ny1 + my2 ⇒ y (m + n) = ny1 + my2 ny1 + my2 ⇒y= ( m + n)
  43. 43. This is called section formula B C ( x, y ) m:n A m A and B in ratio n C is a point on the line joining ( x2 , y 2 ) ( x1 , y1 ) The co-ordinates of C are : nx1 + mx2 x= ( m + n) ny1 + my2 y= ( m + n)
  44. 44. in ratio n C is an exterior point point on the line joining A and B B (x , y ) 2 2 m:n A C ( x, y ) m ( x1 , y1 ) The co-ordinates of C are : nx1 − mx2 x= ( m − n) ny1 − my2 y= ( m − n)
  45. 45. Co-ordinates of circumcentre OF A TRIANGLE WHEN VERTICES ARE GIVEN B( x2 , y2 ) 0 A ( x1 , y1 ) C x1 + x2 + x3 x= 3 ( x2 , y 2 ) y1 + y2 + y3 y= 3
  46. 46. Co-ordinates of in -centre OF A TRIANGLE WHEN VERTICES ARE GIVEN B( x2 , y2 ) c a 0 A ( x1 , y1 ) b C ( x2 , y 2 ) ax1 + bx2 + cx3 ay1 + by2 + cy3 x= y= 3 3
  47. 47. ASSIGNMENT Q1.Which point on x axis is equidistant from (5,9) and (-4,6)? Q2. Which point on y axis is equidistant from (2,3) and (-4,1)?
  48. 48. ASSIGNMENT Q3.Prove that (2a,4a),(2a,6a)and (2a+√3a) are vertices of an equilateral triangle. Q4.In what ratio the x-axis divide the line segment joining the points (2,-3) and (5,6)?
  49. 49. ASSIGNMENT Q5. For what value of x will the points (x,1),(2,1) and (4,5) lie on a line? Q6. Determine the ratio in which the line 3x+y-9=0 divides the segment joining the points (1,3) and (2,7)?
  50. 50. ASSIGNMENT Q7If the points (-2,-1), (1,0),(x,3) and ( 1,y) form a parallelogram, find the value of x and y. Q8.Find the coordinates of (i) centroid ii)incentre (iii)circumcentre of of the triangle whose vertices are (0,6),(8,12) and ( 8,0)
  51. 51. Thank You Developed by Pratima Nayak, Kendriya Vidyalaya,Fort Willia,Kolkata

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