Everything Maths Grade 10

Worked example 9: Points on a straight 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 the three distancesAB,BCandAC

dAB=

√

( 3 0)^2 + (35)^2

=

√

(3)^2 + (2)^2

=

p

9 + 4

=

p

13

dBC=

√

(03)^2 + (57)^2

=

√

(3)^2 + (2)^2

=

p

9 + 4

=

p

13

dAC=

√

((3)3)^2 + (37)^2

=

√

(6)^2 + (4)^2

=

p

36 + 16

=

p

52

Step 3: Find the sum of the two shorter distances

dAB+dBC=

p

13 +

p

13 = 2

p

13 =

p

4 13 =

p

52

Step 4: Explain your answer

dAB+dBC=dAC

therefore pointsA,BandClie on the same straight line.

Exercise 8 – 4:

1.Determine whetherABandCDare parallel, perpendicular or neither if:

a)A(3;4),B(5; 2),C(1;1),D(7; 23)

b)A(3;4),B(5; 2),C(1;1),D(0;4)

c)A(3;4),B(5; 2),C(1; 3),D(2; 2)

304 8.3. Gradient of a line