Question

given that point g is the incenter of △hjk, which of the following is true? gf⊥jf hg = gk hd = hg two of these

Explanation:

Step1: Recall in - center property

The in - center of a triangle is the point of intersection of the angle bisectors of the triangle. The in - center is equidistant from the sides of the triangle. The distance from the in - center to a side of the triangle is measured along a line perpendicular to the side. Since $G$ is the in - center of $\triangle HJK$, the lines from $G$ to the sides of the triangle ($\overline{GF}$, $\overline{GD}$, $\overline{GE}$) are perpendicular to the sides of the triangle. So, $\overline{GF}\perp\overline{JF}$.

Step2: Analyze other options

For option B, $HG$ and $GK$ are not necessarily equal. The in - center is not related to the equality of distances from the in - center to the vertices in the way stated. For option C, $HD$ and $HG$ are not equal. $HD$ is the length of the perpendicular from $H$ to the line containing $G$'s perpendicular to side $HJ$, and $HG$ is the distance from vertex $H$ to the in - center $G$.

Answer:

A. $\overline{GF}\perp\overline{JF}$