Cambridge International Mathematics

260 Coordinate geometry (Chapter 12)

Example 5 Self Tutor

Consider the points A(¡ 2 ,0),B(2,1)and C(1,¡3).

Determine if the triangle ABC is equilateral, isosceles or scalene.

AB=

p

(2¡¡2)^2 +(1¡0)^2

=

p

42 +1^2

=

p

17 units

AC=

p

(1¡¡2)^2 +(¡ 3 ¡0)^2

=

p

32 +(¡3)^2

=

p

18 units

BC=

p

(1¡2)^2 +(¡ 3 ¡1)^2

=

p

(¡1)^2 +(¡4)^2

=

p

17 units

As AB=BC, triangle

ABC is isosceles.

Example 6 Self Tutor

Use the distance formula to show that triangle ABC is right angled

if A is(1,2),Bis(2,5), and C is(4,1).

AB=

p

(2¡1)^2 +(5¡2)^2

=

p

12 +3^2

=

p

10 units

AC=

p

(4¡1)^2 +(1¡2)^2

=

p

32 +(¡1)^2

=

p

10 units

BC=

p

(4¡2)^2 +(1¡5)^2

=

p

22 +(¡4)^2

=

p

20 units

So, AB^2 +AC^2 = 10 + 10 = 20

and BC^2 =20

) triangle ABC is right angled at A:

Example 7 Self Tutor

Findbgiven that A( 3 ,¡ 2 ) and B(b, 1 ) are

p

13 units apart.

From A to B, x-step=b¡ 3

y-step=1¡¡2=3

)

p

(b¡3)^2 +3^2 =

p

13

) (b¡3)^2 +9=13

) (b¡3)^2 =4

) b¡3=§ 2

) b=3§ 2

) b=5or 1 :

A,()-2 ¡0

B,()2 ¡1

C,()1 -3

~` 1 ` 0

~` 1 ` 0

~` 2 ` 0

A

B

C

The right angle

is opposite the

longest side.

There are two

possible solutions in

this example. Draw

a diagram to see

why this is so.

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Y:\HAESE\IGCSE01\IG01_12\260IGCSE01_12.CDR Thursday, 2 October 2008 12:43:39 PM PETER