Lessons Grade 7

Chapter 18

Chapter 18: Coordinate Geometry

18.1 The Cartesian Plane

The coordinate plane is a two-dimensional grid used to locate points. It consists of:

X-axis (horizontal line)

Y-axis (vertical line)

Each point is represented as (x, y).

Example:

The point (3,4) means:

3 steps to the right on the X-axis

4 steps up on the Y-axis

18.2 Distance and Midpoint Formulas

Distance Formula:

d=(x2−x1)2+(y2−y1)2d = sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}d=(x2​−x1​)2+(y2​−y1​)2​

Example:

Find the distance between (1, 2) and (4, 6).

d=(4−1)2+(6−2)2d = sqrt{(4 - 1)^2 + (6 - 2)^2}d=(4−1)2+(6−2)2​d=32+42=9+16=25=5d = sqrt{3^2 + 4^2} = sqrt{9 + 16} = sqrt{25} = 5d=32+42​=9+16​=25​=5

Midpoint Formula:

M=(x1+x22,y1+y22)M = left(frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}right)M=(2x1​+x2​​,2y1​+y2​​)

Example:

Find the midpoint of (2, 8) and (6, 4).

M=(2+62,8+42)=(82,122)=(4,6)M = left(frac{2+6}{2}, frac{8+4}{2}right) = left(frac{8}{2}, frac{12}{2}right) = (4,6)M=(22+6​,28+4​)=(28​,212​)=(4,6)

Real-World Applications:

Coordinate geometry is used in GPS navigation, robotics, video game development, and city planning.

Exercises

Find the distance between (-2,3) and (4,7).

What is the midpoint of (5,2) and (-1,6)?

Plot the points (3,5), (-2,-4), and (0,2) on a coordinate plane.