Everything Maths Grade 10

Worked example 8: Points on a line

QUESTION

Prove thatA(3; 3),B(0; 5)andC(3; 7)are on a straight line.

SOLUTION

Step 1: Draw a sketch

 5  4  3  2  1 1 2 3 4 5

 1

1

2

3

4

5

6

7

A(3; 3)

B(0; 5)

C(3; 7)

x

y

Step 2: Calculate two gradients between any of the three points

m=

y 2 y 1

x 2 x 1

mAB=

5 3

0 (3)

=

2

3

and

mBC=

7 5

3 0

=

2

3

OR

mAC=

3 7

3 3

=

4

6

=

2

3

and

mBC=

7 5

3 0

=

2

3

Step 3: Explain your answer

mAB=mBC=mAC

Therefore the pointsA,BandCare on a straight line.

To prove that three points are on a straight line using the distance formula, we must calculate the distances
between each pair of points and then prove that the sum of the two smaller distances equals the longest
distance.

Chapter 8. Analytical geometry 303