The Coordinate Plane

By the end of this lesson, you will be able to plot and identify points in all four quadrants, calculate distances, and find midpoints.

The Coordinate Plane

y
         |
  II     |    I
  (-,+)  | (+,+)
---------+---------- x
  III    |   IV
  (-,-)  | (+,-)
         |

Two perpendicular number lines divide the plane into four quadrants:

Origin: $(0, 0)$ — where the axes cross.

Ordered pair $(x, y)$: x = horizontal position (right = positive), y = vertical position (up = positive).

Plotting Points

To plot $(3, -2)$:

1. Start at the origin
2. Move right 3 units (x = 3)
3. Move down 2 units (y = -2)
4. Place the point

Quadrant identification:

• Q I: $(+, +)$
• Q II: $(-, +)$
• Q III: $(-, -)$
• Q IV: $(+, -)$

Points on the x-axis have $y = 0$; points on the y-axis have $x = 0$.

Distance Between Two Points

$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

(The Pythagorean theorem applied to the horizontal and vertical distances.)

Example: Distance from $(1, 2)$ to $(4, 6)$:

$d = \sqrt{(4-1)^2+(6-2)^2} = \sqrt{9+16} = \sqrt{25} = 5$

Midpoint

$$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$$

Average the x-coordinates and average the y-coordinates.

Example: Midpoint of $(2, -4)$ and $(6, 2)$:

$M = \left(\frac{2+6}{2}, \frac{-4+2}{2}\right) = (4, -1)$

Summary

Coordinate plane: x-axis (horizontal), y-axis (vertical), origin at $(0,0)$.

Ordered pair $(x,y)$: x = horizontal, y = vertical.

Distance: $d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Midpoint: $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$