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(-2√3, 2√3)

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(-2√3, 2√3)

=  √[(-2√3 + 2)² + (2√3 + 2)²]

=  √(12 + 4 - 8√3 + 12 + 4 + 8√3)

=  √32

BC   = √32

C(-2√3, 2√3) and A(2, 2)

=  √[(2 + 2√3)² + (2 - 2√3)²]

=  √(12 + 4 + 8√3 + 12 + 4 - 8√3)

=  √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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