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?\
\\(\bigcirc\\) a. \\(ad\\)\
\\(\bigcirc\\) b. \\(bd\\)\
\\(\bigcirc\\) c. \\(de\\)\
\\(\bigcirc\\) d. \\(df\\)\
\\(\bigcirc\\) e. \\(ef\\)

Explanation:

The largest circle that fits inside a triangle is its incircle. The intersection of angle bisectors (point D) is the incenter. The radius of the incircle is the perpendicular distance from the incenter to any side of the triangle. Here, $DE$ is the perpendicular segment from incenter D to side $\overline{AC}$, so it equals the incircle radius.

Answer:

C. DE