Question

The right angled triangle circumcenter lies at the _______ of hypotenuse.
A.Inside
B.Outside
C.Midpoint
D.opposite

Answer 1

Hint: Circumcenter is the center of the circle circumscribed on the right angle triangle. A right angle triangle is a triangle whose one angle is ${90}^{\circ }$ . The hypotenuse of the right angled triangle will make ${90}^{\circ }$ angle at any given point on the circumference of the circle. So the hypotenuse of the right triangle will act as a diameter of the circle.

Complete step-by-step answer:
Let PQR be the Right angled triangle in which $\mathrm{\angle }Q={90}^{\circ }$ , PR is the hypotenuse, PQ is the Perpendicular and QR is the base of the triangle. We know that diameter makes a right angle at any point on the circumference of the circle.

We see from the diagram that If Q is a given point on the circumference of the circle then PR is the diameter of the circle as $\mathrm{\angle }Q={90}^{\circ }$. And if the PR is the diameter of the circle then its centre O will be the center of the circle. It is the midpoint of the hypotenuse. So for a right angled triangle, the circumcenter lies at midpoint of the hypotenuse.
Hence the correct answer is C.

Note: Some properties of circumcenter are-
All the vertices of a triangle are at equal distance from the circumcenter.
The circumcenter lies inside the triangle in an acute angled triangle.
The circumcenter outside the triangle is an obtuse angled triangle.