Question

point x is the incenter of $\triangle abc$. if $ex = 4z + 1$, $xf = 2z + 7$, and $m\angle abc = 44^\circ$, find the following measures. $gx = \square$ $m\angle abx = \square^\circ$

Explanation:

Step1: Find z using incenter property

The incenter is equidistant from all sides, so \( EX = XF \).
Set \( 4z + 1 = 2z + 7 \).
Subtract \( 2z \) and \( 1 \): \( 2z = 6 \), so \( z = 3 \).

Step2: Calculate EX (and GX, since incenter is equidistant)

Substitute \( z = 3 \) into \( EX \): \( EX = 4(3) + 1 = 13 \).
Since \( GX = EX \) (incenter equidistant from \( AC \) and \( AB \)), \( GX = 13 \).

Step3: Find \( m\angle ABX \)

The incenter bisects angles, so \( \angle ABX = \frac{1}{2} \angle ABC \).
Given \( \angle ABC = 44^\circ \), \( m\angle ABX = \frac{44^\circ}{2} = 22^\circ \).

Answer:

\( GX = \boldsymbol{13} \), \( m\angle ABX = \boldsymbol{22}^\circ \)