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.

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.

Solved Example on Circumcenter

Find the circumcenter of the triangle in the figure shown.

Choices:
A. (- , - )
B. (, )
C. (, - )
D. (- , )
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 and then find its slope.
Step 3: Midpoint of is (, ) = (, 0)
Step 4: Slope of is - 6.
Step 5: The slope of perpendicular bisector of is the negative reciprocal of - 6, .
Step 6: The perpendicular bisector of passes through the midpoint of .
Step 7: So, the equation of perpendicular bisector of is = implies 2x - 12y = 5.
Step 8: Similarly, the equation of perpendicular bisector of is 8x - 2y = 13.
Step 9: Solving 2x - 12y = 5 and 8x - 2y = 13 gives x = and y = - .
Step 10: So the circumcenter of the given triangle is (, - ).

Related Terms for Circumcenter

  • Center
  • Circle
  • Circumscribed
  • Perpendicular bisector
  • Point of Intersection
  • Polygon
  • Triangle