Question

point a is the incenter of δdef. what is the length of na? options: 2 units, 9 units, 11 units, 14 units. the diagram shows triangle def with incenter a, perpendicular segments from a to sides: ( al = 5x - 1 ) (to ( ed ), right angle at ( l )), ( am = 6x - 3 ) (to ( fd ), right angle at ( m )), and ( an ) (to ( ef ), right angle at ( n )).

Explanation:

Step1: Recall incenter property

The incenter of a triangle is equidistant from all sides, so \( AL = AM = AN \). Thus, \( 5x - 1 = 6x - 3 \).

Step2: Solve for \( x \)

Subtract \( 5x \) from both sides: \( -1 = x - 3 \).
Add 3 to both sides: \( x = 2 \).

Step3: Find \( AL \) (or \( AN \))

Substitute \( x = 2 \) into \( 5x - 1 \): \( 5(2) - 1 = 10 - 1 = 9 \). So \( AN = 9 \).

Answer:

9 units