In this page Circumcentre of triangle question2 we are going to see solution of first question.

**Definition:**

The point of concurrency of the perpendicular bisector of the sides of a triangle is called the circumcentre of the triangle.

**Question 2 :**

Find the co ordinates of the circumcentre of a triangle whose vertices are (0,4) (3,6) and (-8,-2).

Solution :

Let A (0,4), B (3,6) and C (-8,-2) be the vertices of triangle.

Now we need to find the midpoint of the side AB

Midpoint of AB = [(x₁ + x₂)/2 , (y₁ + y₂)/2]

A (0,4) and B (3,6)

Here x₁ = 0, x₂ = 3 and y₁ = 4,y₂ = 6

= [(0 + 3)/2,(4+6)/2]

= [3/2,10/2]

= [3/2,5]

So the vertices of D is (3/2,5)

Slope of AB = [(y₂ - y₁)/(x₂ - x₁)]

= [(6 - 4)/(3 - 0)]

= 2/3

Slope of the perpendicular line through D = -1/slope of AB

= -1/(2/3) ==> -1 x 3/2 ==>-3/2

**Equation of the perpendicular line through D:**

(y-y₁) = m (x-x₁)

Here point D is (3/2,5)

x₁ = 3/2 ,y₁ = 5

(y - 5) = -3/2 (x - 3/2)

y - 5 = -3/4 (2x - 3)

4 (y - 5) = -3 (2x - 3)

4 y - 20 = - 6 x + 9

6 x + 2 y - 20 - 9 = 0

6 x + 2 y - 29 = 0

Equation of the perpendicular line through D is 6 x + 2 y - 29 = 0

Now we need to find the midpoint of the side BC

Midpoint of BC = [(x₁ + x₂)/2 , (y₁ + y₂)/2]

B (3,6) and C (-8,-2)

Here x₁ = 3, x₂ = -8 and y₁ = 6,y₂ = -2

= [(3 + (-8))/2,(6 + (-2))/2]

= [-5/2,4/2]

= [-5/2,2]

So the vertices of E is (-5/2,2)

Slope of BC = [(y₂ - y₁)/(x₂ - x₁)]

= [(-2 - 6)/(-8 - 3)]

= (-8)/(-11)

= 8/11

Slope of the perpendicular line through E = -1/slope of BC

= -1/(8/11) ==> -11/8

**Equation of the perpendicular line through E:**

(y-y₁) = m (x-x₁)

Here point E is (-5/2,2)

x₁ = -5/2 ,y₁ = 2

(y - 2) = -11/8 (x - (-5/2))

(y - 2) = -11/8 (x + 5/2)

(y - 2) = -11/16 (2 x + 5)

16 (y - 2) = -11 (2 x + 5)

16 y - 32 = - 22 x - 55

22 x + 16 y - 32 + 55 = 0

22 x + 16 y + 23 = 0

Equation of the perpendicular line through E is

**22 x + 16 y + 23 = 0 -------(1)**

Now we need to solve the equations of perpendicular bisectors D and E

6 x + 4 y - 29 = 0 ---------(1)

22 x + 16 y + 23 = 0 ---------(2)

(1) x 4 = > 24 x + 16 y - 116 = 0

22 x + 16 y + 23 = 0

(-) (-) (-)

---------------------

2 x - 139 = 0 ==> 2 x = 139 ==> x = 139/2

Substitute x = 139/2 in the (1) equation we get 6 (139/2) + 4 y - 29 = 0

3 (139) + 4 y = 29

417 + 4 y = 29

4 y = 29 - 417

4 y = - 388

y = - 388/4

y = - 97

So the circumcentre of a triangle ABC is **(139/2,-97)**

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