Find the circumcentre of triangle whose
vertices are (-2,3),(2,-1),(4,0)
Answers
Answer:
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
⇒ ( - 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.
Step-by-step explanation: