In this page Circumcentre of triangle question7 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 7:**

**Find the co ordinates of the circumcentre of a triangle whose**

**vertices are ****(-3,-9) (3,9) and (5,-8).**

Let A (-3,-9), B (3,9) and C (5,-8) 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 (-3,-9) and B (3,9)

Here x₁ = -3, x₂ = 3 and y₁ = -9,y₂ = 9

= [(-3+3)/2,(-9+9)/2]

= [0/2,0/2]

= [0,0]

So the vertices of D is (0,0)

Slope of AB = [(y₂ - y₁)/(x₂ - x₁)]

= [(9-(-9))/(3-(-3)]

= (9+9)/(3+3)

= 18/6

= 3

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

= -1/3

__Equation of the perpendicular line through D:__

(y-y₁) = m (x-x₁)

Here point D is (0,0)

x₁ = 0 ,y₁ = 0

(y-0) = -1/3 (x-0)

3 y = -1 (x)

x + 3 y = 0

Equation of the perpendicular line through D is x + 3 y = 0

Now we need to find the midpoint of the side BC

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

B (3,9) and C (5,-8)

Here x₁ = 3, x₂ = 5 and y₁ = 9,y₂ = -8

= [(3 + 5)/2,(9 + (-8)/2]

= [8/2,1/2]

= [4,1/2]

So the vertices of E is (4,1/2)

Slope of BC = [(y₂ - y₁)/(x₂ - x₁)]

= [(-8 - 9)/(5 - 3)]

= -17/2

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

= -1/(-17/2)

= 2/17

__Equation of the perpendicular line through E:__

(y-y₁) = m (x-x₁)

Here point E is (4,1/2)

x₁ = 4,y₁ = 1/2

(y - 1/2) = 2/17 (x - 4)

17(2y - 1)/2 = 2 (x - 4)

34 y - 17 = 4 (x - 4)

34 y - 17 = 4 x - 16

4 x - 34 y - 16 + 17

4 x - 34 y + 1 = 0

4 x - 34 y = -1

Equation of the perpendicular line through E is 4 x - 34 y = -1

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

x + 3 y = 0 ---------(1)

4 x - 34 y = -1 ---------(2)

(1) x 4 => 4 x + 12 y = 0

4 x - 34 y = -1

(-) (+) (+)

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

46 y = 1

y = 1/46

Substitute y = 1/46 in the first equation we get x + 3 y = 0

x + 3 (1/46)= 0

( 46 x + 3) = 0 x 46

46 x + 3 = 0

46 x = -3

x = -3/46

So the circumcentre of a triangle ABC is **(-3/46,1/46) Circumcentre of triangle question7 Circumcentre of triangle question7**

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