Can you help me with answer?

Question

A town has three roads that form an obtuse triangle. James wants to set up a refreshmentsstand so that the shortest distance from his stand to each roads is the sam for all three roads. Where should he set up refreshments stand?

A. On the centroid

B. On the circumcenter

C.on the incenter

D.on the median

Answer 1

All points on the angle bisector of an an angle are equidistant from the rays that make up the angle.

The 3 angle bisectors of a triangle meet at a point called the incenter. The incenter is the center of a circle that meets each of the sides at points of tangency. And the distance from the incenter to each point of tangency is the same as the radius of that circle. So it is the incenter that is equidistant from each of the roads making up the obtuse triangle.

The centroid is the intersection of the 3 medians of the triangle.

The circumcenter is the intersection of the perpendicular bisector of each side. The circumcenter is equidistant from each vertex of the triangle.

Here is a Desmos graph that depicts the 3 radii from the incenter to the points of tangency. You can move the vertices of the triangle to different positions to see that the distances from the incenter to each point of tangency remain the same.

desmos.com/calculator/7gllsj1p2u

Here is a graph depicting the circumcenter of a triangle along with the corresponding circumcircle.

desmos.com/calculator/zrjgxurqcp

Another graph showing centroid as intersection of medians.

desmos.com/calculator/5ps3j6ppq2

Finally here is a graph that shows the intersection of the 3 altitudes of a triangle (orthocenter).

desmos.com/calculator/0kk4zu6spq