Question

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

• Analyzing the first option: Angle bisectors meet at the incenter, not the orthocenter. So this is incorrect.
• Analyzing the second option: If \(\overline{BE} \perp \overline{AC}\), \(\overline{AG} \perp \overline{BC}\), and \(\overline{CF} \perp \overline{AB}\), then \(BE\), \(AG\), and \(CF\) are altitudes of the triangle. The intersection of altitudes is the orthocenter, so this fits the definition of an orthocenter.
• Analyzing the third option: Segments that bisect the sides (medians) meet at the centroid, not the orthocenter. So this is incorrect.
• Analyzing the fourth option: Perpendicular bisectors of the sides meet at the circumcenter, not the orthocenter. So this is incorrect.

Answer:

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