Lesson 2: Solving Problems on the Coordinate Plane

Property

For any two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$:

• If $x_1 = x_2$, the points lie on the same vertical line.
• If $y_1 = y_2$, the points lie on the same horizontal line.
• The distance from a point $(x, y)$ to the x-axis is $y$.
• The distance from a point $(x, y)$ to the y-axis is $x$.

Examples

• Given points $A(2, 5)$ and $B(6, 3)$, point $A$ is farther from the x-axis because its y-coordinate, $5$, is greater than the y-coordinate of point $B$, which is $3$. Point $B$ is farther from the y-axis because its x-coordinate, $6$, is greater than the x-coordinate of point $A$, which is $2$.
• The points $C(4, 1)$, $D(4, 3)$, and $E(4, 6)$ all lie on the same vertical line because they share the same x-coordinate, $x=4$.
• The points $F(1, 8)$ and $G(5, 8)$ both lie on the same horizontal line because they share the same y-coordinate, $y=8$.

Explanation

The coordinates of a point tell you its exact position relative to the origin and the axes. The x-coordinate indicates the horizontal distance from the y-axis, while the y-coordinate indicates the vertical distance from the x-axis. By comparing the x- and y-coordinates of different points, you can determine their relative positions without graphing. Points sharing the same x-coordinate are aligned vertically, and points sharing the same y-coordinate are aligned horizontally.

Property

To find the distance between two points on the same horizontal or vertical line, find the difference of their non-identical coordinates.

Examples

• On a city map, the library is at $(2, 5)$ and the school is at $(2, 9)$. Since they are on the same vertical line, the distance between them is $9 - 5 = 4$ blocks.
• A treasure map shows a pirate ship at $(4, 3)$ and a hidden treasure at $(6, 3)$. Since they are on the same horizontal line, the distance between them is $6 - 4 = 2$ paces.
• A zoo map marks the entrance at $(4, 4)$ and the lion exhibit is 2 units left and 1 unit down. Since left subtracts from the $x$-coordinate and down subtracts from the $y$-coordinate, the lion exhibit is at $(4-2, 4-1) = (2, 3)$.

Explanation

The coordinate plane can be used to represent real-world locations and solve problems involving distance. An ordered pair $(x, y)$ can describe the exact position of an object on a map or grid. To find the distance between two points that share a coordinate, you can simply count the units between them or subtract their coordinates that are different. This method works for finding lengths along straight horizontal or vertical paths.