Geometry

Question Analysis

The main focus is to determine which type of triangle has an orthocenter outside the triangle by understanding the properties of altitudes in different triangles.

Key Concept Explanation

The orthocenter is the point where the altitudes of a triangle intersect. An altitude is a perpendicular line segment from a vertex to the opposite side.

In an acute triangle, all angles are less than 90°, and the altitudes intersect inside the triangle.

In a right triangle, one angle is 90°, and the orthocenter is at the vertex of the right angle.

In an obtuse triangle, one angle is greater than 90°. The lines containing the altitudes will intersect outside the triangle.

Step - by - Step Solution

1. Analyze each type of triangle:

Acute triangle: Since all angles are less than 90°, the altitudes intersect inside the triangle. So the orthocenter is inside, not outside.

Obtuse triangle: When one angle is greater than 90°, the lines of the altitudes (extended in some cases) will cross outside the triangle. Thus, the orthocenter is outside.

Right triangle: With a 90° angle, th...