Introduction to coordinate geometry by pratima nayak
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The document discusses distance formula and section formula to find the coordinates of points on a plane. It explains how to find the distance between two given points using the distance formula. It also describes how to find the coordinates of a point C that divides the line segment between points A and B in a given ratio using the section formula. The document provides examples of finding coordinates of circumcenter and incenter of a triangle given its vertices. It concludes with some assignment questions related to these concepts.
Introduction to coordinate geometry by pratima nayak
- 1. How much is the distance between flower and the butterfly ?
- 7. 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
- 8. 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
- 9. 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
- 10. 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
- 11. 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
- 12. 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
- 13. 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
- 14. 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
- 16. 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
- 18. -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
- 20. -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
- 22. 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
- 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. 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
- 25. 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
- 26. 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
- 27. 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
- 28. (3,3) Let there is a butterfly at (1,1) (1,1) (0,0)
- 29. How much is the distance between flower and the butterfly ? (1,1) (3,3)
- 30. 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).
- 31. 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.
- 32. 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
- 33. B (x , y ) 2 2 A ( x1 , y1 ) O How much is the distance between the points A and B ?
- 34. 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.
- 35. 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
- 36. 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
- 37. 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
- 38. 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
- 39. B (x , y ) 2 2 C is a point on the line joining A and B in ratio m:n A ( x1 , y1 ) O
- 40. 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?
- 41. 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
- 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 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)
- 43. 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)
- 44. 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)
- 45. 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)
- 46. 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
- 47. 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
- 48. 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)?
- 49. 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)?
- 50. 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)?
- 51. 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)
- 52. Thank You Developed by Pratima Nayak, Kendriya Vidyalaya,Fort Willia,Kolkata