Question

in abc, the angle bisectors meet at point d. point e is on $overline{ac}$, and $overline{de}$ is perpendicular to $overline{ac}$. point f is the location where the perpendicular bisectors of the sides of the triangle meet. what is the radius of the largest circle that can fit inside abc?

a. ad
b. bd
c. de
d. df
e. ef

Explanation:

Step1: Recall the in - center property

The largest circle that can fit inside a triangle is the incircle. The in - center of a triangle is the point of intersection of the angle bisectors of the triangle. Here, point D is the in - center as the angle bisectors meet at point D.

Step2: Recall the radius of the incircle

The radius of the incircle is the perpendicular distance from the in - center to any of the sides of the triangle. Since DE is perpendicular to side AC and D is the in - center, DE is the radius of the incircle.

Answer:

C. DE