# HOW TO SHOW THE GIVEN POINTS FORM AN ISOSCELES TRIANGLE OR EQUILATERAL TRIANGLE

Example 1 :

Show that the following points taken in order form an isosceles triangle.

A(5, 4), B(2, 0), C (–2, 3)

Solution :

Distance between two points  =  √[(x2 - x1)2 + (y2 - y1)2]

√[(2 - 5)2 + (0 - 4)2]

√[(-3)2 + (-4)2]

√(9 + 16)

√25

AB  =  5

B(2, 0), C(–2, 3)

=  √[(-2 - 2)2 + (3 - 0)2]

=  √[(-4)2 + 32]

=  √(16 + 9)

=  √25

BC  =  5

C(–2, 3) and A(5, 4)

=  √[(5 -(-2))2 + (4 - 3)2]

=  √[(5+2)2 + 12]

=  √(49 + 1)

=  √50

CA  =  5√2

The sides AB and BC are having equal length. So, it is an isosceles triangle.

Example 2 :

Show that the following points taken in order form an isosceles triangle.

A(6, –4), B(–2, –4), C(2, 10)

Solution :

Distance between two points  =  √[(x2 - x1)2 + (y2 - y1)2]

A(6, –4), B(–2, –4)

=  √[(-2 - 6)2 + (-4 + 4)2]

=  √[(-8)2 + (0)2]

=  √64

AB  =  8

B(–2, –4), C(2,10)

=  √[(2 + 2)2 + (10 + 4)2]

=  √(42 + 142)

=  √(16 + 196)

=  √212

BC   = 2√53

C(2,10)  A(6,–4)

=  √[(6 - 2)2 + (-4 - 10)2]

=  √[42 + (-14)2]

=  √(16 + 196)

=  √212

CA   = 2√53

The sides BC and CA are having equal length. Hence it is an isosceles triangle.

Example 3 :

Show that the following points taken in order form an equilateral triangle.

A(2, 2), B(–2, –2), C(-23, 23)

Solution :

A(2, 2), B(–2, –2)

=  √[(-2 - 2)2 + (-2 - 2)2]

=  √(-4)2 + (-4)2

=  √16 + 16

AB  =  √32

B(–2, –2), C(-23, 23)

=  √[(-23 + 2)2 + (23 + 2)2]

=  √(12 + 4 - 83 + 12 + 4 + 83)

=  √32

BC   = √32

C(-23, 23) and A(2, 2)

=  √[(2 + 23)2 + (2 - 23)2]

=  √(12 + 4 + 83 + 12 + 4 - 83)

=  √32

CA   = √32

The sides AB, BC and CA are having equal length.Hence it is equilateral triangle.

Example 4 :

Show that the following points taken in order form an equilateral triangle.

A(3, 2), B(0, 1), C(0, 3)

Solution :

A(3, 2), B(0, 1)

=  √[(0 - 3)2 + (1 - 2)2]

=  √[(-3)2 + (-1)2]

=  √(3 + 1)

AB  =  √4

AB  =  2

B (0, 1), C(0, 3)

=  √[(0 - 0)2 + (3 - 1)2]

=  √4

BC   =  2

C (0, 3) and A (3, 2)

=  √[(3 - 0)2 + (2 - 3)2]

=  √(3 + 1)

=  √4

CA   = 2

The sides AB, BC and CA are having equal length.Hence it is equilateral triangle.

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