Question

show that the points (12,8) (-2,6) and (6, 0) are the vertices of a isosceles right angled triangle​

Answer 1

Step-by-step explanation:

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Explanation:

Explanation:Given vertices A(12,8),B(−2,6), &C(6,0).

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Answer 2

Answer:

The given points are vertices of isosceles right triangle.

Step-by-step-explanation:

Let the triangle be ABC.

• A ( 12, 8 ) ≡ ( x₁, y₁ )
• B ( - 2, 6 ) ≡ ( x₂, y₂ )
• C ( 6, 0 ) ≡ ( x₃, y₃ )

Now, by distance formula,

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

⇒ d ( A, B ) = √{ [ 12 - ( - 2 ) ]² + ( 8 - 6 )² }

⇒ d ( A, B ) = √[ ( 12 + 2 )² + 2² ]

⇒ d ( A, B ) = √[ ( 14 )² + 4 )

⇒ d ( A, B ) = √( 196 + 4 )

⇒ d ( A, B ) = √200

∴ Length of AB = √200 units

∴ AB² = ( √200 )²

⇒ AB² = 200 units - - - ( 1 )

Now,

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

⇒ d ( B, C ) = √[ ( - 2 - 6 )² + ( 6 - 0 )² ]

⇒ d ( B, C ) = √[ ( - 8 )² + ( 6 )² ]

⇒ d ( B, C ) = √( 64 + 36 )

⇒ d ( B, C ) = √( 100 )

⇒ d ( B, C ) = 10

∴ Length of BC = 10 units - - - ( 2 )

Now,

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

⇒ d ( A, C ) = √[ ( 12 - 6 )² + ( 8 - 0 )² ]

⇒ d ( A, C ) = √( 6² + 8² )

⇒ d ( A, C ) = √( 36 + 64 )

⇒ d ( A, C ) = √( 100 )

⇒ d ( A, C ) = 10

∴ Length of AC = 10 units - - - ( 3 )

From ( 2 ) & ( 3 )

Length of BC = Length of AC

∴ △ABC is an isosceles triangle.

Now,

BC² + AC² = ( 10 )² + ( 10 )²

⇒ BC² + AC² = 100 + 100

⇒ BC² + AC² = 200 units - - - ( 4 )

From ( 1 ) & ( 4 ),

BC² + AC² = AB²

By converse of Pythagoras theorem,

m∠C = 90°

∴ △ABC is an isosceles right triangle.

∴ The given points are vertices of isosceles right triangle.

Hence shown!