Question

When constructing a circle inscribed in a triangle, what is the purpose of constructing a perpendicular segment from the incenter to a side of the triangle?

To determine the center of the circle
To determine the radius of the circle
To connect the midpoints of each side of the triangle
To connect the arc markings from the angle bisectors

Answer 1

Please refer to the picture I've attached.

When a circle is inscribed in a triangle, it means that the circle is drawn inside a triangle such that the midpoints of each side of the triangle are tangent to the circle. By doing this, you draw the biggest circle that could fit inside a triangle.

Now, the circle and the triangle have the same center illustrated here by the red dot. When you connect all sides of the triangle to the center (yellow lines), you create three equal angles. This is equal to 360 (one revolution) divided by 3, equals 120°. But when you create a perpendicular bisector from the center (black line), the angle is halved. It will now be 120/2 = 60°. Usually you are given the sides of the triangle. By using pythagorean theorems, you can find, therefore, the radius of the circle. To illustrate, the radius is equal to (s/2)/tan60.

Thus, the answer is: To determine the radius of the circle.