Consider the triangle ABC with vertices A (-2,3), B (2,1) and C (0,2)Q. What is the circumcentre of the triangle ABC?a)(-2,-2)b)(2,2)c)(-2,2)d)(2,-2)Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Circumcentre of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. To find the circumcentre of triangle ABC, we need to find the equations of the perpendicular bisectors of AB, BC, and AC, and then find the point of intersection.

Finding the midpoint of AB:
The midpoint M of AB can be found using the midpoint formula:
M = ((x1 + x2)/2, (y1 + y2)/2)
M = ((-2 + 2)/2, (3 + 1)/2)
M = (0, 2)

Finding the slope of AB:
The slope of AB can be found using the slope formula:
m = (y2 - y1)/(x2 - x1)
m = (1 - 3)/(2 - (-2))
m = -2/4
m = -1/2

Finding the perpendicular slope of AB:
The perpendicular slope of AB is the negative reciprocal of the slope of AB, which is 2.
So, the perpendicular slope of AB is 2.

Finding the equation of the perpendicular bisector of AB:
Using the point-slope form of a linear equation, we can write the equation of the perpendicular bisector of AB:
(y - y1) = m(x - x1)
(y - 2) = 2(x - 0)
y - 2 = 2x
2x - y = -2

Similarly, we can find the equations of the perpendicular bisectors of BC and AC:
Perpendicular bisector of BC: 2x + y = 4
Perpendicular bisector of AC: x + 2y = 5

Finding the point of intersection:
To find the circumcentre, we need to find the point of intersection of the three perpendicular bisectors.
Solving the system of equations formed by the three equations of the perpendicular bisectors, we find that the point of intersection is (-2, -2).

Therefore, the circumcentre of triangle ABC is option A (-2, -2).

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