Question

show that the points (3,3) , ( -2,-2) and (8, -2) are the vertices of an isosceles right-angled triangle​

Answer 1

Answer:

The given points are vertices of isosceles right triangle.

Step-by-step-explanation:

Let the triangle be ABC.

• A ( 3, 3 ) ≡ ( x₁, y₁ )
• B ( - 2, - 2 ) ≡ ( x₂, y₂ )
• C ( 8, - 2 ) ≡ ( x₃, y₃ )

Now, by distance formula,

d ( A, B ) = √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ]

⇒ d ( A, B ) = √{ [ 3 - ( - 2 ) ]² + [ 3 - ( - 2 ) ]² }

⇒ d ( A, B ) = √[ ( 3 + 2 )² + ( 3 + 2 )² ]

⇒ d ( A, B ) = √( 5² + 5² )

⇒ d ( A, B ) = √( 25 + 25 )

⇒ d ( A, B ) = √50

∴ Length of AB = √50 units - - - ( 1 )

Now,

d ( B, C ) = √[ ( x₂ - x₃ )² + ( y₂ - y₃ )² ]

⇒ d ( B, C ) = √{ ( - 2 - 8 )² + [ - 2 - ( - 2 ) ]² }

⇒ d ( B, C ) = √[ ( - 10 )² + ( - 2 + 2 )² ]

⇒ d ( B, C ) = √( 100 + 0² )

⇒ d ( B, C ) = √100

⇒ d ( B, C ) = 10

∴ Length of BC = 10 units

∴ BC² = ( 10 )²

⇒ BC² = 100 units - - - ( 2 )

Now,

d ( A, C ) = √[ ( x₁ - x₃ )² + ( y₁ - y₃ )² ]

⇒ d ( A, C ) = √{ ( 3 - 8 )² + [ 3 - ( - 2 ) ]² }

⇒ d ( A, C ) = √[ ( - 5 )² + ( 3 + 2 )² ]

⇒ d ( A, C ) = √( 25 + 5² )

⇒ d ( A, C ) = √( 25 + 25 )

⇒ d ( A, C ) = √50

∴ Length of AC = √50 units - - - ( 3 )

From ( 1 ) & ( 3 )

Length of AB = Length of AC

∴ △ABC is an isosceles triangle.

Now,

AB² + AC² = ( √50 )² + ( √50 )²

⇒ AB² + AC² = 50 + 50

⇒ AB² + AC² = 100 units - - - ( 4 )

From ( 2 ) & ( 4 ),

AB² + AC² = BC²

By converse of Pythagoras theorem,

m∠A = 90°

∴ △ABC is an isosceles right triangle.

∴ The given points are vertices of isosceles right triangle.

Hence shown!