Question

based on the given diagram, complete the sentence below.

diagram of triangle abc with point d, and perpendicular segments af, bg, ce

answer attempt 1 out of 2

point d is the ______ of △abc because \\(\overline{af}\\), \\(\overline{bg}\\), and \\(\overline{ce}\\) are all...

Explanation:

Step1: Analyze the diagram elements

In the triangle \( \triangle ABC \), segments \( \overline{AF} \), \( \overline{BG} \), and \( \overline{CE} \) are all altitudes (perpendicular to the opposite sides, as indicated by the right angles at \( F \), \( G \), and \( E \)).

Step2: Recall triangle centroid/orthocenter definitions

The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. Since \( \overline{AF} \), \( \overline{BG} \), and \( \overline{CE} \) are altitudes (perpendicular to sides) and intersect at \( D \), \( D \) is the orthocenter. (Note: If they were medians, it would be centroid, but here they are altitudes with right angles, so orthocenter.)

Answer:

orthocenter; altitudes (completing the sentence: Point \( D \) is the orthocenter of \( \triangle ABC \) because \( \overline{AF} \), \( \overline{BG} \), and \( \overline{CE} \) are all altitudes.)