Question

find the coordinates of circumcenter and radius of circumcircle of triangle ABC

Answer 1

Solution:

Given points are,

A = (3, 2), B = (1, 4), C = (5, 4)

To find out the circumcenter we have to solve any two bisector equations and find out the intersection points.

So, mid point of AB = (3+12,2+42) = (2,3)

Slope of AB = (4−21−3) = -1

Slope of the bisector is the negative reciprocal of the given slope.

So, the slope of the perpendicular bisector = 1

Equation of AB with slope 1 and the coordinates (2,3) is,

(y – 3) = 1(x – 2)

x – y = -1………………(1)

Similarly, for AC

Mid point of AC = (3+52,2+42) = (4,3)

Slope of AC = (4−25−3) = 1

Slope of the bisector is the negative reciprocal of the given slope.

So, the slope of the perpendicular bisector = -1

Equation of AC with slope -1 and the coordinates (4,3) is,

(y – 3) = -1(x – 4)

y – 3 = -x + 4

x + y = 7………………(2)

By solving equation (1) and (2),

(1) + (2) ⇒ 2x = 6; x = 3

Substitute the value of x in to (1)

3 – y = -1

y = 3 + 1 = 4

So the circumcenter is (3, 4)