Circumcenter

Definition of Circumcenter

Circumcenter of a triangle is the point of intersection of all the three perpendicular bisectors of the triangle.

More About Circumcenter

    The circle drawn around the triangle by taking circumcenter as the center is called a circumscribed circle.

Video Examples: Circumcenters


Example of Circumcenter

    In the above diagram, the three perpendicular bisectors PO, QO, and RO of sides BC, AB, and AC of the triangle ABC intersect at the point O. So, the point O is called the circumcenter of the triangle ABC.
      examples of Circumcenter

Solved Example on Circumcenter

Ques: Find the circumcenter of the triangle in the figure shown.

    example of Circumcenter
    Choices:
    A. (-73/46,-7/46)
    B. (73/46,7/46)
    C. (73/46,-7/46)
    D. (-73/46,7/46)
    Correct Answer: C

Solution:

    Step 1: The point where all the perpendicular bisectors intersect is called circumcenter.
    Step 2: To find the perpendicular bisector of , find the midpoint of example of Circumcenter and then find its slope.
    Step 3: Midpoint of example of Circumcenter is ((2+3)/2,(3-3)/2 ) = (5/2, 0)
    Step 4: Slope of example of Circumcenter is - 6
    Step 5: The slope of perpendicular bisector of example of Circumcenter is the negative reciprocal of - 6,1/6
    Step 6: The perpendicular bisector of example of Circumcenter passes through the midpoint of example of Circumcenter
    Step 7: So, the equation of perpendicular bisector of example of Circumcenter isexample of Circumcenter = 1/6 implies 2x - 12y = 5.
    Step 8: Similarly, the equation of perpendicular bisector of example of Circumcenter is 8x - 2y = 13.
    Step 9: Solving 2x - 12y = 5 and 8x - 2y = 13 gives x =73/46 and y = -7/46 .
    Step 10: So the circumcenter of the given triangle is (73/46, -7/46 )