Topic 10 of 12

The coordinate plane

Step 8 · Ratios and negative numbers topics

Locate points with ordered pairs on a grid with four quadrants, reflect points across the axes, and find distances and areas using coordinates.

Two number lines that cross

The coordinate plane is made of two number lines. The horizontal one is the x-axis and the vertical one is the y-axis. They cross at the origin, which is 0 on both. Right of the origin and above it are positive; left and below are negative.

Ordered pairs

A point is named by an ordered pair $(x, y)$ . The first number says how far to go right (positive) or left (negative) from the origin. The second says how far up (positive) or down (negative). The order matters: $(3, 2)$ and $(2, 3)$ are different points.

Example: Plotting a point

Where is the point $(-4, 3)$ ?

Start at the origin.
The x value is $-4$ , so go 4 units left.
The y value is 3, so go 3 units up.

Answer4 left and 3 up from the origin.

Quadrants

The axes divide the plane into four quadrants, numbered counter-clockwise starting at the top right.

Quadrant	Where	Signs of $(x, y)$	Example
I	top right	$(+, +)$	$(3, 5)$
II	top left	$(-, +)$	$(-3, 5)$
III	bottom left	$(-, -)$	$(-3, -5)$
IV	bottom right	$(+, -)$	$(3, -5)$

A point with a 0 in it lies on an axis and is in no quadrant. $(0, 5)$ is on the y-axis and $(-2, 0)$ is on the x-axis.

Reflections

Reflecting a point across an axis flips it to the other side, the same distance away. Reflecting across the x-axis changes the sign of y. Reflecting across the y-axis changes the sign of x.

Example: Reflecting

Reflect $(3, -2)$ across the x-axis, and then across the y-axis.

Across the x-axis, the sign of y changes: $(3, -2)$ becomes $(3, 2)$ .
Across the y-axis, the sign of x changes: $(3, -2)$ becomes $(-3, -2)$ .

Answer $(3, 2)$ across the x-axis, and $(-3, -2)$ across the y-axis.

Distance between points

When two points have the same x or the same y, they lie on a straight horizontal or vertical line, and the distance is found with absolute value. If they are on opposite sides of an axis, add the distances from it. If they are on the same side, subtract.

Example: A vertical distance

How far apart are $(-4, 3)$ and $(-4, -5)$ ?

The x values match, so the points are on a vertical line.
3 is 3 above the x-axis and $-5$ is 5 below it: $|3| + |-5| = 8$ .

Answer8 units.

Example: A horizontal distance

How far apart are $(-3, 1)$ and $(4, 1)$ ?

The y values match, so the points are on a horizontal line.
$-3$ is 3 left of the y-axis and 4 is 4 right: $3 + 4 = 7$ .

Answer7 units.

Shapes on the plane

Joining points gives a shape, and distances give its sides.

Example: A rectangle

A rectangle has corners $(1, 1)$ , $(5, 1)$ , $(5, 4)$ and $(1, 4)$ . Find its perimeter and area.

The bottom side runs from $(1, 1)$ to $(5, 1)$ : length $5 - 1 = 4$ .
The right side runs from $(5, 1)$ to $(5, 4)$ : length $4 - 1 = 3$ .
Perimeter: $2 \times (4 + 3) = 14$ . Area: $4 \times 3 = 12$ .

AnswerPerimeter 14 units and area 12 square units.

Try it yourself

Work each one on paper first, then open the answer to check.

In which quadrant is $(-5, 2)$ ?
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Quadrant II, because x is negative and y is positive.
In which quadrant is $(4, -7)$ ?
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Quadrant IV, because x is positive and y is negative.
Reflect $(-2, 5)$ across the x-axis.
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$(-2, -5)$ . The sign of y changes.
Find the distance between $(-6, 2)$ and $(3, 2)$ .
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9, because they are on opposite sides of the y-axis: $6 + 3 = 9$ .
Find the distance between $(0, -3)$ and $(0, 8)$ .
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11, because $3 + 8 = 11$ .
A rectangle has corners $(-2, -1)$ , $(3, -1)$ , $(3, 2)$ and $(-2, 2)$ . Find its area and perimeter.
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The width is $2 + 3 = 5$ and the height is $1 + 2 = 3$ . The area is 15 square units and the perimeter is 16 units.