Coordinate Geometry , Distance and Midpoint

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.

Descartes, a French philosopher and mathematician, introduced the basic concepts of coordinate system that still bears his name, the Cartesian coordinate system. René Descartes (1596 - 1650)

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.

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

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

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).

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

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)

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)

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)

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

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

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

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:

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

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

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

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