Question

Find the circum centre of the triangle whose vertices are (-2, -3), (-1, 0) and (7, -6) also find circum

radius.​

Answer 1

Answer:

The coordinates of the circumcentre of the triangle are P ( 27, 29 ).

The circumradius of the triangle is 43.19 units.

Step-by-step-explanation:

We have given the coordinates of the vertices of a triangle.

We have to find the coordinates of the circumcentre and the circumradius of the triangle.

Let the vertices of the triangle be A, B and C respectively and P be the circumcentre of the triangle.

A ≡ ( - 2, - 3 ) ≡ ( x₁, y₁ )

B ≡ ( - 1, 0 ) ≡ ( x₂, y₂ )

C ≡ ( 7, - 6 ) ≡ ( x₃, y₃ )

P ≡ ( x, y )

Now, we know that,

The circumcentre of a triangle is equidistant from each vertex of the triangle.

∴ PA = PB = PC

⇒ PA² = PB² = PC² - - - ( 1 ) [ Squaring each term ]

Now,

PA² = PB² - - - [ From ( 1 ) ]

⇒ { √[ ( x - x₁ )² + ( y - y₁ )² ] }² = { √[ ( x - x₂ )² + ( y - y₂ )² ] }² - - - [ Distance formula ]

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

⇒ [ x - ( - 2 ) ]² + [ y - ( - 1 ) ]² = [ x - ( - 1 ) ]² + ( y - 0 )²

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

⇒ x² + 4x + 4 + y² + 2y + 1 = x² + 2x + 1 + y² - - - [ ( a + b )² = a² + 2ab + b² ]

⇒ 4x + 4 + 2y + 1 = 2x + 1 - - - [ Cancelling x² & y² from both sides ]

⇒ 4x + 2y - 2x = 1 - 4 - 1

⇒ 4x - 2x + 2y = 1 - 1 - 4

⇒ 2x + 2y = 0 - 4

⇒ 2x + 2y = - 4

⇒ x + y = - 2 - - - [ Dividing both sides by 2 ]

⇒ x = - 2 + y

⇒ x = y - 2 - - - ( 2 )

Now,

PB² = PC² - - - [ From ( 1 ) ]

⇒ { √[ ( x - x₂ )² + ( y - y₂ )² ] }² = { √[ ( x - x₃ )² + ( y - y₃ )² ] }² - - - [ Distance formula ]

⇒ ( x - x₂ )² + ( y - y₂ )² = ( x - x₃ )² + ( y - y₃ )²

⇒ [ x - ( - 1 ) ]² + ( y - 0 )² = ( x - 7 )² + [ y - ( - 6 ) ]²

⇒ ( x + 1 )² + y² = ( x - 7 )² + ( y + 6 )²

⇒ x² + 2x + 1 + y² = x² - 14x + 49 + y² + 12y + 36 - - - [ ( a ± b )² = a² ± 2ab + b² ]

⇒ 2x + 1 = - 14x + 49 + 12y + 36 - - - [ Canceling x² & y² from both sides ]

⇒ 2x + 14x - 12y = 49 + 36 - 1

⇒ 16x - 12y = 49 + 35

⇒ 16x - 12y = 49 + 30 + 5

⇒ 16x - 12y = 79 + 5

⇒ 16x - 12y = 84

⇒ 4x - 3y = 21 - - - ( 3 ) [ Dividing both sides by 4 ]

⇒ 4 * ( y - 2 ) - 3y = 21 - - - [ From ( 2 ) ]

⇒ 4y - 8 - 3y = 21

⇒ 4y - 3y = 21 + 8

⇒ y = 29

By substituting y = 29 in equation ( 2 ), we get,

x = y - 2 - - - ( 2 )

⇒ x = 29 - 2

⇒ x = 27

∴ The coordinates of the circumcentre of the triangle are P ( 27, 29 ).

────────────────────────

Now, we know that,

Circumradius of a triangle is the distance between the circumcentre and any of the vertices of the triangle.

∴ PA, PB and PC are circum radii of the triangle.

∴ d ( P, A ) = √[ ( x - x₁ )² + ( y - y₁ )² ] - - [ Distance formula ]

⇒ d ( P, A ) = √{ [ 27 - ( - 2 ) ]² + [ 29 - ( - 3 ) ]² }

⇒ d ( P, A ) = √[ ( 27 + 2 )² + ( 29 + 3 )² ]

⇒ d ( P, A ) = √[ ( 29 )² + ( 32 )² ]

⇒ d ( P, A ) = √( 841 + 1024 )

⇒ d ( P, A ) = √1865

⇒ d ( P, A ) = 43.185

⇒ d ( P, A ) = 43.19 units ( Approx. )

∴ The circumradius of the triangle is 43.19 units.