Question

The point equidistant from the three sides of a triangle is

A. Circumcentre

B. Centroid

C. Incentre

D. Orthocentre

Answer 1

Hint: Let us plot circumcentre, centroid, incentre and orthocentre for the triangle ABC and then look at which point is equidistant from the three sides of the triangle.

Complete Step-by-Step solution:

Let ABC be a triangle.

As we know that the circumcenter of a polygon is the center of the circle that contains all the vertices of the polygon, if such a circle exists. For a triangle, it always has a unique circumcenter and thus a unique circumcircle.

And circumcenter of the triangle ABC is plotted below

So, as we can see from the above figure, the circumcentre of the triangle is not equidistant from all its sides.

The centroid of a triangle is the intersection of the three medians, or the "average" of the three vertices.

And the median is a line from one vertex of the triangle to the center of the side opposite to that.

And centroid of the triangle ABC is plotted below

So, as we can see from the above figure that the circumcenter of the triangle is not equidistant from all its sides.

The centroid of a triangle is the intersection of the three medians, or the "average" of the three vertices.

And the median is a line from one vertex of the triangle to the center of the side opposite to that.

And centroid of the triangle ABC is plotted below

So, as we can see from the above figure, the centroid of the triangle is not equidistant from all its sides.

Now as we know that the angle of the triangle is the center of the inscribed circle and we know that the inscribed circle is the circle that is inside the triangle and touches all its sides.

As we know that the center of the circle is equidistant from all points that lie on the circle.

So, the center of the inscribed circle (incentre will be equidistant from all sides of the triangle.

So, the incentive is plotted below.

So, as we can see from the above figure that the incentre of the triangle is equidistant from all its sides.

And the orthocenter of the triangle is the intersection of the three altitudes of the triangle.

And altitude of the triangle is the line drawn perpendicular from the vertex of a triangle to the side opposite to that vertex.

So, the orthocenter of triangle ABC is plotted below.

So, as we can see from the above figure that the orthocenter of the triangle is not equidistant from all its sides.
Hence, the correct option will be C.

Note: Whenever we come up with this type of problem then first, we assume a triangle ABC. And then using the definition of circumcenter, incentre, centroid and orthocenter plot a triangle ABC. Then we can clearly see from that figure which point is equidistant from all its sides. And this will be the easiest and most efficient way to find the solution to the problem.