Lesson 4: Polygons in the Coordinate Plane

Property

The Standard Coordinatization of the Plane
Draw a horizontal line and a vertical line, so that the point of intersection is center page. That point is called the origin and is denoted $(0,0)$.
Decide which variable is to be measured along the horizontal line ($x$), and which is measured along the vertical line ($y$).
Select a measure to represent a unit on both axes. Now, $(x, y)$ is found by moving $x$ units on the horizontal axis, and then $y$ units in the vertical direction.

Examples

• To plot the point $(2, 5)$, start at the origin $(0,0)$, move 2 units to the right, and then 5 units up. This point is in Quadrant 1.
• The point $(-4, -1)$ is found by moving 4 units to the left from the origin and then 1 unit down. This point is in Quadrant 3.
• A point with a zero coordinate lies on an axis. The point $(0, 3)$ is on the y-axis, and the point $(-5, 0)$ is on the x-axis.

Explanation

The coordinate plane is like a map made from two number lines (axes) that cross at a right angle. The point $(x, y)$ tells you how far to move horizontally (x-value) and then vertically (y-value) from the start (origin).

Session 2. Finding Horizontal and Vertical Side Lengths

Property

To find the length of a side that is perfectly horizontal or vertical, you can either count the grid spaces between the points or find the difference between their coordinates:

• Horizontal lines: Subtract the x-coordinates (the y-coordinates stay the same).
• Vertical lines: Subtract the y-coordinates (the x-coordinates stay the same).

Examples

• The distance between (3, 6) and (10, 6) is a horizontal line. The length is 10 - 3 = 7 units.
• The distance between (4, 2) and (4, 9) is a vertical line. The length is 9 - 2 = 7 units.
• A rectangle has vertices at (1, 1), (6, 1), (6, 5), and (1, 5). Its width (horizontal) is 6 - 1 = 5 units, and its height (vertical) is 5 - 1 = 4 units.