Question

Find the third vertices of an equilateral triangle of vertices are (1,1) and (-1,-1).

Answer 1

Answer:

The coordinates of the third vertex of the triangle are either ( - √3, √3 ) or ( √3, - √3 ).

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

Let the equilateral triangle be △ABC.

• A ≡ ( 1, 1 ) ≡ ( x₁, y₁ )
• B ≡ ( - 1, - 1 ) ≡ ( x₂, y₂ )
• C ≡ ( x, y )

We know that,

All the three sides of an equilateral triangle are equal in length.

∴ AB = BC = AC

⇒ d ( A, B ) = d ( B, C ) = d ( A, C )

Now,

d ( A, B ) = d ( B, C )

By distance formula,

√[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ] = √[ ( x₂ - x )² + ( y₂ - y )² ]

By squaring both sides, we get,

⇒ ( x₁ - x₂ )² + ( y₁ - y₂ )² = ( x₂ - x )² + ( y₂ - y )²

⇒ [ 1 - ( - 1 ) ]² + [ 1 - ( - 1 ) ]² = ( - 1 - x )² + ( - 1 - y )²

⇒ ( 1 + 1 )² + ( 1 + 1 )² = ( - 1 )² - 2 * ( - 1 ) * x + x² + ( - 1 )² - 2 * ( - 1 ) * y + y²

⇒ ( 2 )² + ( 2 )² = 1 + 2x + x² + 1 + 2y + y²

⇒ 4 + 4 = 1 + 1 + 2x + x² + 2y + y²

⇒ 8 = 2 + 2x + x² + 2y + y²

⇒ x² + 2x + y² + 2y + 2 - 8 = 0

⇒ x² + 2x + y² + 2y - 6 = 0 - - - ( 1 )

Now,

d ( B, C ) = d ( A, C )

By distance formula,

√[ ( x₂ - x )² + ( y₂ - y )² ] = √[ ( x₁ - x )² + ( y₁ - y )² ]

By squaring both sides, we get,

⇒ ( x₂ - x )² + ( y₂ - y )² = ( x₁ - x )² + ( y₁ - y )²

⇒ ( - 1 - x )² + ( - 1 - y )² = ( 1 - x )² + ( 1 - y )²

⇒ ( - 1 )² - 2 * ( - 1 ) * x + x² + ( - 1 )² - 2 * ( - 1 ) * y + y² = ( 1 - x )² + ( 1 - y )²

⇒ 1 + 2x + x² + 1 + 2y + y² = ( 1 )² - 2 * 1 * x + x² + ( 1 )² - 2 * 1 * y + y²

⇒ 1 + 1 + 2x + 2y + x² + y² = 1 - 2x + x² + 1 - 2y + y²

⇒ 1 + 1 + 2x + 2y + x² + y² = 1 + 1 - 2x - 2y + x² + y²

⇒ 2x + 2y = - 2x - 2y

⇒ 2x + 2x = - 2y - 2y

⇒ 4x = - 4y

⇒ x = - y

By substituting x = - y in equation ( 1 ), we get,

x² + 2x + y² + 2y - 6 = 0 - - - ( 1 )

⇒ ( - y )² + 2 * ( - y ) + y² + 2y - 6 = 0

⇒ y² - 2y + y² + 2y - 6 = 0

⇒ y² + y² - 2y + 2y - 6 = 0

⇒ 2y² - 6 = 0

⇒ 2y² = 6

⇒ y² = 6 ÷ 2

⇒ y² = 3

⇒ y = ± √3

Now,

x = - y

⇒ x = - ( ± √3 )

⇒ x = ∓ √3

∴ ( x, y ) = ( ∓ √3, ± √3 )

∴ The coordinates of the third vertex of the triangle are either ( - √3, √3 ) or ( √3, - √3 ).