How do you find the circumcenter of a triangle given the coordinates?

Question

     I have to find the circumcenter of the triangle:
B(2,2)
P(4,10)
H(12,2)
Can someone walk me through it? Please be detailed because I get confused easily... I hate geometry....

Answer 1

Circumecenter: the point where perpendicular bisectors of a triangle meet  called circumcenter.

BP,BH,PH are the sides of given tringle.

Step1Find the mid pionts of each line

B(2,2),  P(4,10), H(12,2)

Mid point of BP = (2+4/2,2+10/2)

= (3,6)

Midpoint of PH = (4+12/2, 10+2/2)

= (8,6)

Midpoint of HB = (12+2/2, 2+2/2)

= (7,2)

Step2 find slope of the each line say m.

Slope of BP m== 10-2/4-2 = 4

Slope of PH m = 2-10/12-4=-1

Slope of HB m = 2-2/12-2 = 0

Step3 find the slope of the perpendicular bisectors of BP,PH,HB 

Slope of the bisector lines = -1/slope of the line

Slope of bisector line of BP say m1= -1/4

slope of bisector line of PH say m2 =-1/-1

= 1

Slope of the bisector line of HB = -1/0 is not defined.

Step4 find bisector line equations using midpoints(3,6)and (8,6).

m1=-1/4 ,m2 = 1

bisector line of BP is y-6 =-1/4(x-3)

4(y-6) = -1(x-3)

4y-24 = -x+3

4y-24+x-3 = 0

x+4y = 27  --------->(1)

Bisector line of  PH is y-6 = 1(x-8)

y-6 = x-8

y-6-x+8 = 0

-x+y  = -2  ----------> (2)

To find the x,y values Add equation (1) and (2)

x+4y = 27

-x+y = -2

________

5y = 25

y = 25/5 = 5

Substitute the y value in (2)

-x+5 = -2

-x = -2-5

-x =- 7

Cancell the signs each side.

x = 7

There fore circumcenter of the triangle is (7,5)