The axis is a ruler
On an axis, the distance between two points is the difference of their readings.
Two points on the x-axis
A(4, 0) and B(6, 0) are two points on the x-axis. O is the origin, so OA = 4 units and OB = 6 units.
How far apart are A and B?
What is the distance AB?
What this lesson covers
The idea
The distance between two points on the x-axis is the difference of their x-coordinates (AB = OB – OA), and between two points on the y-axis it is the difference of their y-coordinates.
Two points on the x-axis
A(4, 0) and B(6, 0) are two points on the x-axis. O is the origin, so OA = 4 units and OB = 6 units.
How far apart are A and B?
What is the distance AB?
- 10 units
- 2 units
- 24 units
Slide points along the axes
A and B sit on the x-axis, C and D on the y-axis. The blue bar runs from O to the nearer point. The amber bar is the gap between the two points. Drag a point, or choose it and press the arrows.
Subtract the readings
AB = OB − OA. The distance between two points on the x-axis is the difference of their x-coordinates. On the y-axis it is the difference of their y-coordinates.
A(4, 0) and B(6, 0): AB = 6 − 4 = 2 units. C(0, 3) and D(0, 8): CD = 8 − 3 = 5 units.
Take the bigger reading minus the smaller one, so a distance is never negative. It works with negative coordinates too: for P(−3, 0) and Q(2, 0), PQ = 2 − (−3) = 5.
Notes
AB = OB − OA. On the x-axis the distance between two points is the difference of their x-coordinates. On the y-axis it is the difference of their y-coordinates.
Check yourself
A(2, 0) and B(9, 0) lie on the x-axis. Find AB.
Answer: 7 units
AB = 9 − 2 = 7 units. On the ruler, it is 7 steps from 2 to 9.
C(0, −2) and D(0, 6) lie on the y-axis. Find CD.
Answer: 8 units
CD = 6 − (−2) = 6 + 2 = 8 units. From −2 to 0 is 2 units, and from 0 to 6 is 6 units.
P is at (5, 0) on the x-axis. Q is also on the x-axis, and PQ = 3. Where can Q be?
A(4, 0) is on the x-axis and C(0, 3) is on the y-axis. Can you find AC by subtracting, 4 − 3?
- (2, 0) only. Q can be 3 units to the left of P, at (2, 0). But it can also be 3 units to the right.
- (2, 0) or (8, 0) — correct. Yes! 5 − 3 = 2 and 5 + 3 = 8. Both points are 3 units from P.
- (8, 0) only. Q can be 3 units to the right of P, at (8, 0). But it can also be 3 units to the left.
- Yes. AC = 4 − 3 = 1.. Subtracting works only for two points on the same axis. Here AC slants across the plane, and it is longer than 3 or 4.
- No. The points are on different axes, so AC is a slanting line. — correct. Yes! You will see in the next lesson how Pythagoras measures it. (AC turns out to be 5 units.)