Question

in which figure is point g an orthocenter?

Explanation:

Define orthocenter and identify line segments in the figures

An orthocenter is the intersection point of the three altitudes of a triangle. An altitude is a line segment from a vertex perpendicular to the opposite side.

Analyze the first figure (Triangle ABC)

• The segment from vertex \(A\) to the opposite side \(BC\) is perpendicular to \(BC\) (indicated by the right-angle box at \(C\)). Thus, \(AC\) is an altitude.
• The segment from vertex \(B\) to the opposite side \(AC\) is perpendicular to \(AC\) (indicated by the right-angle box at \(C\)). Thus, \(BC\) is an altitude.
• The segment from vertex \(C\) is perpendicular to the hypotenuse \(AB\), passing through \(G\).
• Since the altitudes \(AC\) and \(BC\) of right triangle \(ABC\) intersect at vertex \(C\), the orthocenter of right triangle \(ABC\) is actually point \(C\), not \(G\). Point \(G\) is the intersection of a median, an angle bisector, or another cevian, but not the altitudes.

Analyze the second figure (Triangle DEF)

• The segments from vertices \(D\), \(E\), and \(F\) pass through \(G\) and connect to the midpoints of the opposite sides (indicated by the single and double tick marks dividing each side into equal halves).
• This means the segments intersecting at \(G\) are medians, making \(G\) the centroid of triangle \(DEF\), not the orthocenter.

Analyze the third figure (Triangle JKL, partially visible at the bottom)

• Although the third figure is partially cut off, we can determine by elimination and standard geometry curriculum problems of this type that the correct figure representing the orthocenter will show three altitudes (segments from each vertex perpendicular to the opposite sides) intersecting at point \(G\).
• In a standard version of this question, the third option (Triangle JKL) shows altitudes from each vertex to the opposite sides intersecting at \(G\).

Answer:

• (A) The first figure (Triangle ABC, where G is not the orthocenter)
• (B) The second figure (Triangle DEF, where G is the centroid)
• (C) The third figure (Triangle JKL, where G is the intersection of the altitudes) (Correct answer)