1. Distance between Any Two Points on a Plane

2. Well done. Now, try to find the distance between A and B.
Do you remember how to calculate the distance between P and Q? x y
P( , 5) 1
Q( , 5)
B( , 5) 5
A( , 2) 1
AB is neither a horizontal line nor a vertical line. I don’t know how to calculate the distance.
The distance between P and Q is (5 
1) units
= 4 units.

3. Distance between Any Two Points on a Plane
BC is a vertical line.
Consider two points A(1, 2) and B(5, 5) on a rectangular coordinate plane. 2 3 4 1 x y 5
B(5, ) 5
Draw a horizontal line from A and
Coordinates of C =
( , ) 5 2
a vertical line from B.
AC = (5 – 1) units =
4 units
4 units
The two lines intersect at C. C
( , ) 1 5 2
BC = (5 – 2) units =
3 units
3 units
A( , 2)
By Pythagoras’ theorem, 2 + = BC AC AB
AC is a horizontal line.
units 3 4 2 + =
units 5 =
5 units

4. It is known as the distance formula between two points.
In general, for any two points A(x1, y1) and B(x2, y2) on a rectangular coordinate plane, x y
A(x1, y1)
B(x2, y2)
y2 – y1
C( , ) x2 y1
x2 – x1 AB ( ) 2 1 y x - + =
It is known as the distance formula between two points.

5. Find the length of PQ in the figure.y
Find the length of PQ in the figure.
Q( , ) 9 6
P( , ) 3 1 x = PQ [9 -
3)] ( - 2 + (6 - 1) 2
units
Remember to write the ‘units’. = 12 2 + 5 2
units =
144 + 25
units
units
169 =
units 13 =

6. Follow-up question
In each of the following, find the distance between the two given points.
(a) A(2, 1) and B(5, 5) (b) C(1, 2) and D(7, 6)
(Leave your answers in surd form if necessary.)
Solution
(a) = AB ) 2 5 ( - 2 + ) 1 5 ( - 2
units
units 16 9 + =
units 25 =
units 5 =

7. Follow-up question
In each of the following, find the distance between the two given points.
(a) A(2, 1) and B(5, 5) (b) C(1, 2) and D(7, 6)
(Leave your answers in surd form if necessary.)
Solution
(b) = CD 1) 7 ( - 2 +
2)] ( [6 -
units 2
units 8 8) ( 2 + - =
units 64 + =
units) 2 8
(or
units
128 =