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Coordinate Geometry , Distance and Midpoint

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1 Coordinate Geometry , Distance and Midpoint

2 Learning Objectives: Demonstrate an understanding of Coordinate Plane and the related terms. Find the distance between two points in the plane. Find the coordinates of the midpoint.

3 Descartes, a French philosopher and mathematician, introduced the basic concepts of coordinate system that still bears his name, the Cartesian coordinate system. René Descartes ( )

4 Coordinate plane A coordinate plane is a plane containing two perpendicular number lines, one horizontal and the other vertical, meeting each other at zero. The common point is called the origin, denoted by O.

5 The horizontal number line is called X-axis and the vertical number line is called Y-axis.

6 The axes uniquely determine the position of every point in a coordinate plane.
B(-4, 3) A(3, 2) C(-2, -3)

7 Coordinates The ordered pair of numbers that gives the location of a point on a Cartesian plane is called the coordinates of that point. Generally, any point is written in the form P(x, y) where, P is the name of the point, x is called the x-coordinate (abscissa) and y is called the y-coordinate (ordinate).

8 Practice Question Identify the coordinates of the points marked on following Cartesian plane. B K A D C O J I H E L G F

9 Plotting points on a plane
Every ordered pair of numbers will uniquely determine a point on a Cartesian plane. To plot a point (x, y), you have to start from origin and move x units to the right (if x is positive) or left (if x is negative) and from there move y units up (if y is positive) or down (if y is negative). The new location is exactly the point (x, y)

10 Example 1: To plot a point (2, -3), you have to start from origin and move 2 units to the right and from there move 3 units down. The new location is exactly the point (2, -3) (2, -3)

11 Example 2: Plot the following points on a Cartesian plane: A(2, 3)
B(-2, 1.5) C(1, -3) D(-4, -3) E(0, 2.5) F(-4, 0) G(1.2, π) H(0, -3.5) I(2, 0) J(-2, -2) O(0, 0)

12 Quadrants The coordinate axes will divide the plane into four regions called the quadrants.

13 Example 3 Fill in the following table with the quadrants in which the given points lie: Point Quadrant A(1, -5) IV B(-2, -4) III C(3, 5) I D(-3, 1.5) II E(0, -5) Y-axis F(1, 0) X-axis

14 The Distance Formula (x2,y2) d (x1,y1) The distance d between the points (x1,y1) and (x2,y2) is :

15 Example 4: Find the distance between the two points.
(-2,5) and (3,-1) Let (x1,y1) = (-2,5) and (x2,y2) = (3,-1) Solution:

16 Example 5: Find the distance between the two points.
(-3,4) and (0,0) Solution:

17 The Midpoint Formula (x2,y2) M (x1,y1) The midpoint M of a line segment joining two points (x1,y1) and (x2,y2) is:

18 Example 6: Find the midpoint of the segment whose endpoints are (6,-2) & (2,-9)
Solution:

19 Example 7: Find the midpoint of the line segment whose endpoints are (-8,-10) and the origin. Solution:


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