Question

given that point g is the circumcenter of △abc, which of the following is true? segment ag is congruent to segment gc. segment bf is congruent to segment ec. segment dg is perpendicular to segment gf. two of these are true.

Explanation:

Step1: Recall circum - center property

The circum - center of a triangle is the point of intersection of the perpendicular bisectors of the sides of the triangle. It is equidistant from the vertices of the triangle.

Step2: Analyze segment AG and GC

Since G is the circum - center of $\triangle ABC$, by the property of the circum - center, $AG = BG=CG$. So segment AG is congruent to segment GC.

Step3: Analyze segment BF and EC

There is no information indicating that segment BF is congruent to segment EC. Just because G is the circum - center, we cannot conclude this.

Step4: Analyze segment DG and GF

There is no information given to suggest that segment DG is perpendicular to segment GF. Just because G is the circum - center, we cannot make this conclusion.

Answer:

A. Segment AG is congruent to segment GC.