Question

prove that the points are the vertices of a right angled isosceles triangle (1, 2), (1, 5) and (4, 2)

Answer 1

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Answer 2

Therefore the given vertices of the triangle form the' Right-angled Isosceles Triangle'.

Given:

The vertices of a triangle: ( 1,2 ), ( 1,5 ), and ( 4,2 )

To Find:

The given vertices form a right-angled isosceles triangle or not.

Solution:

The given question can be solved very easily as shown below.

Given vertices of a triangle: ( 1,2 ), ( 1,5 ), and ( 4,2 )

Let Point A = ( 1,2 )

Point B = ( 1,5 )

Point C = ( 4,2 )

Let us find the distance between the points.

⇒ Distance between 2 points ( x₁, y₁ ) and ( x₂, y₂ ) = √( x₂ - x₁ )² + ( y₂ - y₁ )²

⇒ Distance between A and B = AB = √ ( 1 - 1 )² + ( 5 - 2 )²

⇒ Distance between A and B = AB = √3² = 3 units

⇒ Distance between B and C = BC = √ ( 4 - 1 )² + ( 2 - 5 )²

⇒ Distance between B and C = BC = √ 3² + ( -3 )² = 3√2 units

⇒ Distance between C and A = CA = √ ( 1 - 4 )² + ( 2 - 2 )²  

⇒ Distance between C and A = CA = √ ( -3 )² = 3 units

Applying Pythagoras Theorem,

⇒ BC² = AB² + CA²

⇒ ( 3√2 )² = 3² + 3²

⇒ 9 × 2 = 9 + 9

⇒ 18 = 18

⇒ LHS = RHS

So given triangle is 'Right-angled'.

And also AB = CA = 3 units.

So the triangle is 'Isosceles'.

Therefore the given vertices of the triangle form the' Right-angled Isosceles Triangle'.

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