Question

a is the incenter of $\triangle pqr$. find $au$.

Explanation:

Step1: Recall Incenter Property (Angle Bisector)

The incenter \( A \) of a triangle is the intersection of angle bisectors. So, \( \angle TPA = \angle UPA \), meaning \( 3x + 2 = 4x - 9 \).

Step2: Solve for \( x \)

\[

$$\begin{align*} 3x + 2 &= 4x - 9 \\ 2 + 9 &= 4x - 3x \\ x &= 11 \end{align*}$$

\]

Step3: Recall Incenter Property (Equal Distances to Sides)

The incenter is equidistant from all sides of the triangle. So, \( AT = AU = AK \). Given \( AT = 20 \), then \( AU = 20 \).

Answer:

\( 20 \)