Question

Find the cicumcentre of triangle(i) (-2,3),(2,-1),(4,0) ​

Answer 1

Answer:

(3/2,5/2) is circumcentre.

Step-by-step explanation

Let O (x, y) be the circumcentre of the triangle.

Then O A = OB = O C

If we take O A = OB

Using distance formula

O A ² = OB²

⇒ (x + 2) ² + (y - 3) ² = (x - 2) ² + (y + 1) ²

⇒ x² + 4 + 4 x + y² + 9 - 6 y = x²+ 4 - 4 x + y² + 1 + 2 y

After cancelling x² and y² from both sides we get linear equation

8 x - 8 y +8 = 0

Or x – y +1 = 0

⇒ y = x + 1 ……………………(1)

Now consider OC = OB

O C ² = OB²

⇒ (x - 4) ² +(y - 0) ² = (x - 2) ² + (y + 1) ²

⇒ x² + 16 - 8 x + y² = x² + 4 - 4 x + y² + 1 +2 y

After cancelling x² and y² from both sides we get linear equation

- 4 x + 11 - 2 y = 0 ………….(2)

Substituting value of y from (1) equation to equation(2)

- 4 x + 11 - 2 (x + 1) = 0

X = 3/2

Since y = x + 1= 3/2 + 1 = 5/2

So Circumcentre is O (3/2 , 5/2)

Check:

O A^2 = (3/2+2) ² +(5/2-3) ² =49/4 + ¼ = 50/4=25/2

O B^2 = (3/2-2) ² + (5/2+1) ²= ¼ + 49/4 = 50/4 = 25/2

O C^2 = (3/2-4) ² +(5/2-0) ² =25/4 +25/4 = 50/4 =25/2

So clearly O(3/2 , 5/2) is equidistant from A,B and C.

Hence O(3/2,5/2) is cicumcentre.