Chapter 7: Coordinates

7.1 Distance in a Cartesian Coordinate System

The Cartesian coordinate system consists of two perpendicular axis lines: the horizontal $x$-axis and the vertical $y$-axis, intersecting at the origin $O(0, 0)$.

Distance Between Two Points

• Horizontal Line (parallel to $x$-axis): If $y_1 = y_2$, the distance between $(x_1, y_1)$ and $(x_2, y_2)$ is: $$\text{Distance} = |x_2 - x_1|$$
• Vertical Line (parallel to $y$-axis): If $x_1 = x_2$, the distance between $(x_1, y_1)$ and $(x_2, y_2)$ is: $$\text{Distance} = |y_2 - y_1|$$
• Slanted Line (General Case): Using Pythagoras' theorem, the distance $d$ between any two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

7.2 Midpoint in the Cartesian Coordinate System

The midpoint is the point that lies exactly halfway along the line segment joining two given points.

Midpoint Formula

The coordinates of the midpoint $M$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are given by the average of their $x$-coordinates and $y$-coordinates:

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

7.3 The Coordinate System

The coordinate system allows us to solve geometric problems, analyze shapes on a grid, calculate perimeters, and determine missing coordinates of geometric figures such as parallelograms, rectangles, and triangles.