Question

in the diagram, which must be true for point d to be an orthocenter?
be, cf, and ag are angle bisectors.
be ⊥ ac, ag ⊥ bc, and cf ⊥ ab.
be bisects ac, cf bisects ab, and ag bisects bc.
be is a perpendicular bisector of ac, cf is a perpendicular bisector of ab, and ag is a perpendicular bisector of bc.

Explanation:

The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. So, we need to check which option describes three segments that are altitudes (perpendicular from a vertex to the opposite side).

• The first option describes angle bisectors, which meet at the incenter, not orthocenter.
• The second option: $\overline{BE} \perp \overline{AC}$ (so BE is an altitude from B to AC), $\overline{AG} \perp \overline{BC}$ (altitude from A to BC), and $\overline{CF} \perp \overline{AB}$ (altitude from C to AB). This matches the definition of altitudes, whose intersection is the orthocenter.
• The third option describes medians (segments from vertex to midpoint of opposite side), which meet at the centroid.
• The fourth option describes perpendicular bisectors, which meet at the circumcenter.

So the correct option is the one with the three perpendicular segments as altitudes.

Answer:

B. $\overline{BE} \perp \overline{AC}$, $\overline{AG} \perp \overline{BC}$, and $\overline{CF} \perp \overline{AB}$