Question
Triangle A B C is shown. Lines are drawn from each point to the opposite side and intersect at point D. They form line segments A G, B E, and C F.
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.
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.
Answers
GPT-4o mini
For point D to be considered the orthocenter of triangle ABC, the lines drawn from each vertex (A, B, C) to the opposite side must be the altitudes of the triangle. This means that these lines should be perpendicular to the sides they intersect.
Therefore, the correct condition that must hold true for point D to be the orthocenter is:
**BE ⊥ AC, AG ⊥ BC, and CF ⊥ AB.**
This indicates that line segment BE from vertex B to side AC is perpendicular to AC, line segment AG from vertex A to side BC is perpendicular to BC, and line segment CF from vertex C to side AB is perpendicular to AB.
Therefore, the correct condition that must hold true for point D to be the orthocenter is:
**BE ⊥ AC, AG ⊥ BC, and CF ⊥ AB.**
This indicates that line segment BE from vertex B to side AC is perpendicular to AC, line segment AG from vertex A to side BC is perpendicular to BC, and line segment CF from vertex C to side AB is perpendicular to AB.