Chapter 7 · The coordinate plane
Distances Along Horizontal and Vertical Lines
Find how far apart two points are on a horizontal or vertical line. Then find a point that is a given distance left, right, up or down from another point.
You can already graph a point from its ordered pair. Now you will use ordered pairs to measure how far apart two points are.
The distance between two points is the length of the straight path from one point to the other. You measure it in units. On every grid in this lesson, one grid step is unit.
This lesson measures only two kinds of straight path:
- A horizontal path runs left and right. Two points are on the same horizontal line when they have the same -coordinate.
- A vertical path runs up and down. Two points are on the same vertical line when they have the same -coordinate.
On these paths, the distance is the number of units you travel along the line. You can count the grid steps, or you can work the distance out from the coordinates. Do not judge it by how long the gap looks in a picture, because a picture can be drawn at any size.
All three points in the figure above have the -coordinate , so they are on the same horizontal line. Only their -coordinates differ: , and .
Point B to Point C. Count the grid steps from B to C. There are . Both points are to the right of the -axis. Point C is units from the axis and Point B is units from it. The path from the axis to C passes through B. Take away the units from the axis to B, and the part from B to C is left: units.
Point A to Point B. Count the grid steps from A to B. There are . These points are on opposite sides of the -axis, so the path between them crosses it. The path covers units from A to the axis, then more units from the axis to B: units.
Two questions decide each distance on this line. How far is each point from the axis? Are the points on the same side of it or on opposite sides?
Key concept
Distance between points that share a coordinate
This rule is only for two points that share a coordinate. If they share a -coordinate, they are on a horizontal line: use their -coordinates. If they share an -coordinate, they are on a vertical line: use their -coordinates.
Take the absolute value of each coordinate you are using. It is that point's distance from an axis: the -axis for an -coordinate, the -axis for a -coordinate.
- One positive and one negative: the points are on opposite sides of that axis. Add the absolute values.
- Both positive or both negative: the points are on the same side. Subtract the smaller absolute value from the larger.
If one of these coordinates is , that point is on the axis, and adding or subtracting gives the same distance.
Worked example
Two points on the same side of an axis
Find the distance between and .
Compare the ordered pairs. Both x-coordinates are −5, so the points are on the same vertical line and the rule applies. On a vertical line the y-coordinates are the ones that differ, so use them: −2 and −8.
Take the absolute value of each y-coordinate. This gives each point's distance from the x-axis.
Both y-coordinates are negative, so both points are below the x-axis, on the same side. The path from the axis to the farther point passes through the nearer point. Subtract the smaller distance from the larger to keep only the part between the two points.
Answer
The distance between and is units.
Worked example
Two points on opposite sides of an axis
Find the distance between and .
Both y-coordinates are −1, so the points are on the same horizontal line and the rule applies. Use the x-coordinates, −7 and 2, because they are the ones that differ.
Take the absolute value of each x-coordinate to get each point's distance from the y-axis.
One x-coordinate is negative and the other is positive, so the points are on opposite sides of the y-axis. The path between them crosses the axis, so it includes both distances. Add them.
Answer
The distance between and is units.
Now turn the question around. You know one point and a distance, and you need to find another point.
Start at the known point and move that many units straight left, right, up or down. A move left or right stays on the same horizontal line, so only the -coordinate changes. A move up or down stays on the same vertical line, so only the -coordinate changes. The other coordinate stays exactly as it was.
On the horizontal line through the known point, two points are that far away: one to the left and one to the right. On the vertical line through it, there are two more: one above and one below. This lesson looks for such points only on those two lines.
To find the coordinate that changes, use the point's distance from an axis: the -axis for a move left or right, the -axis for a move up or down. For a starting point that is not on that axis, there are two cases.
- A move away from the axis leaves the point farther from it on the same side. Add the move to the distance from the axis. The sign of the coordinate, positive or negative, stays the same.
- A move toward the axis first uses the units needed to reach it. A shorter move stops before the axis: subtract it from the distance, and the sign stays the same. A longer move crosses the axis: the leftover units are the new distance from the axis on the other side, where the sign changes. A move of exactly that length lands on the axis, where the coordinate is .
The figure below shows Point D at and four points that are each units from it: Point E to the left, Point F to the right, Point G above and Point H below. The worked example after the figure finds their coordinates.
Worked example
Points 5 units from a given point
Point D is at . Find the points that are units from Point D on the horizontal line through it and on the vertical line through it.
Find how far Point D is from each axis. Its x-coordinate is −2, so it is 2 units left of the y-axis. Its y-coordinate is 1, so it is 1 unit above the x-axis.
Move 5 units left to reach Point E. This move goes away from the y-axis, so add the 5 units to the 2 units. A point 7 units left of the y-axis has x-coordinate −7. The y-coordinate stays 1, so Point E is at (−7, 1).
Move 5 units right to reach Point F. This move goes toward the y-axis. The first 2 units reach the axis, so subtract to find the units left over. Those 3 units go past the axis to its right side, where x-coordinates are positive. The y-coordinate stays 1, so Point F is at (3, 1).
Move 5 units up to reach Point G. This move goes away from the x-axis, so add the 5 units to the 1 unit. A point 6 units above the x-axis has y-coordinate 6. The x-coordinate stays −2, so Point G is at (−2, 6).
Move 5 units down to reach Point H. This move goes toward the x-axis. The first 1 unit reaches the axis, and the 4 units left over go below it, where y-coordinates are negative. The x-coordinate stays −2, so Point H is at (−2, −4).
Check by measuring back to Point D. Measuring back undoes each move, so it should return 5 units. Each new point shares a coordinate with Point D, so the distance rule applies. Subtract when both coordinates are positive or both are negative. Add when one is positive and one is negative.
Answer
On the horizontal line through Point D, and are each units from it. On the vertical line through Point D, and are each units from it.
Here is the idea to keep. When two points share an -coordinate or a -coordinate, the distance between them comes from the two coordinates that differ. Use absolute value to find how far each point is from the axis. Then add if the points are on opposite sides of the axis, or subtract the smaller from the larger if they are on the same side.
To find a point at a given distance, work the other way. Move that many units along the horizontal or vertical line, change only one coordinate, and check whether the move crosses an axis.
Next, you will use a coordinate plane as a map, explain why absolute value gives the distance between points, and spot mistakes in finding distance.