Question

which best explains why the orthocenter of an obtuse triangle is outside the triangle?
all three of the altitudes lie entirely outside the triangle.
two of the altitudes lie entirely outside the triangle.
all three of the medians lie entirely outside the triangle.
two of the medians lie entirely outside the triangle.

Explanation:

Step1: Recall orthocenter and obtuse triangle

In an obtuse triangle, one angle is greater than \(90^\circ\). The orthocenter is the intersection of altitudes. For an obtuse triangle, two altitudes are drawn from the vertices of the acute angles to the opposite sides (which are extended), so these two altitudes lie outside the triangle. The third altitude (from the obtuse angle vertex) lies inside the triangle.

Step2: Analyze options

• Option A: All three altitudes outside? No, one is inside.
• Option B: Two of the altitudes lie entirely outside? Yes, matches the property.
• Option C: Medians? Medians always lie inside the triangle, so this is wrong.
• Option D: Medians? Medians are inside, so wrong.

Answer:

B. Two of the altitudes lie entirely outside the triangle.