Question

point x is the in - center of δabc. if ex = 4z + 1, xf = 2z + 7, and m∠abc = 44°, find the following measures. gx = m∠abx = °

Explanation:

Step1: Use property of in - center

The in - center of a triangle is equidistant from the sides of the triangle. So, $EX = XF$.
$4z + 1=2z + 7$

Step2: Solve for z

Subtract $2z$ from both sides: $4z-2z + 1=2z-2z + 7$, which gives $2z+1 = 7$. Then subtract 1 from both sides: $2z+1 - 1=7 - 1$, so $2z=6$. Divide both sides by 2: $z = 3$.

Step3: Find GX

Since $EX = XF=GX$ (in - center property), substitute $z = 3$ into the expression for $EX$ or $XF$. Using $XF=2z + 7$, we have $XF=2\times3 + 7=6 + 7 = 13$. So $GX = 13$.

Step4: Find $m\angle ABX$

The in - center is the point of intersection of the angle bisectors of a triangle. Since $BX$ is an angle bisector of $\angle ABC$ and $m\angle ABC = 44^{\circ}$, then $m\angle ABX=\frac{1}{2}m\angle ABC$. So $m\angle ABX=\frac{1}{2}\times44^{\circ}=22^{\circ}$.

Answer:

$GX = 13$
$m\angle ABX = 22$