Altitudes
Overview
An altitude of a triangle is the perpendicular segment from a vertex to the line containing the opposite side. Unlike a median, an altitude is defined by a right angle, not by a midpoint — and in an obtuse triangle the altitude from an acute vertex lands outside the triangle, on the extension of the opposite side. The three altitudes meet at the orthocenter. In an isosceles triangle, the altitude from the vertex angle does extra duty: it is also the median, the angle bisector, and the perpendicular bisector of the base.
Vocabulary
- Altitude of a Triangle
- A segment that goes from a vertex of a triangle, perpendicular to the line containing the opposite side.
- Perpendicular
- Intersecting at a right angle (90°).
- Concurrent
- Three or more lines or segments that intersect at a single point.
- Orthocenter
- The point where the three altitudes of a triangle meet.
Main Ideas
- An altitude is perpendicular to the line containing the opposite side, so it forms a 90° angle where it lands.
- An altitude can appear outside the triangle — in an obtuse triangle, the altitude from an acute vertex falls on the extension of the opposite side.
- In a right triangle, both legs are altitudes, and the orthocenter is the vertex of the right angle.
- A triangle has up to three altitudes, concurrent at the orthocenter.
- In an isosceles triangle, the perpendicular bisector, angle bisector, median, and altitude from the vertex angle to the base are all the same segment; in an equilateral triangle this is true from any vertex.
- Because an altitude is perpendicular, its slope is the opposite reciprocal of the slope of the side it meets.
Lesson Notes
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Lesson Video
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Practice Problems
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