Question

In the following figures, are the centre of the circle and the circumcentre of the triangle the same?

• A. Yes, Yes
• B. No, Yes
• C. No, No
• D. Yes, No

Answer 1

The correct option is A

Yes, Yes

We take $\Delta$ABC and draw the perpendicular bisector of any of two sides.

XY and MN are respectively the perpendicular bisector of sides AB and BC. They meet at a point O.

Now if we join the point OA, OB, OC

Let P be the point on AB from which the perpendicular bisector of AB passes.

In $\Delta$OPA and $\Delta$OPB

$\angle$OPB = $\angle$OPA = 90º

Side OP is common to both triangles.

AP = BP (as side AB is bisected by line XY)

Therefore by SAS congruency the $\Delta$OPA and $\Delta$OPB are congruent.

Thus OA = OB.

Similarly OB = OC.

Hence OA = OB = OC.

Thus we can make circle through points A, B, C with Centre O. ___________________ (1)

Now, if we take arc ABC and join the points. We make the perpendicular bisector of line segments AB and BC then find the points at which they are intersecting. We find that the perpendicular bisector meet at point O. __________________ (2)

From conclusion (1) and (2), we conclude that the circumcentre of these triangle is same as the centre of circle.

Similarly we can say for the other figure that the circumcentre of the triangle is the centre of the circle