Altitudes & Orthocenter
An altitude is the perpendicular line from a vertex to the opposite side (or its extension). The three altitudes meet at the orthocenter.
Introduction
The orthocenter is the last of the four major triangle centers. Unlike the centroid (always inside), the orthocenter can be inside, on, or outside the triangle — depending on whether it's acute, right, or obtuse.
Past Knowledge
Perpendicular lines (3.1). Triangle centers (6.1.2–6.1.4).
Today's Goal
Identify altitudes, locate the orthocenter, and compare all 4 centers.
Future Success
Euler line. Area via altitudes. Coordinate geometry applications.
Key Concepts
Altitude
A perpendicular segment from a vertex to the line containing the opposite side. Unlike a median, it need not hit the midpoint and may land outside the triangle.
Orthocenter
The point where all three altitudes meet.
- Acute triangle: orthocenter is inside
- Right triangle: orthocenter is at the vertex of the right angle
- Obtuse triangle: orthocenter is outside
Summary: 4 Triangle Centers
| Center | Formed By | Key Property |
|---|---|---|
| Circumcenter | ⊥ bisectors of sides | Equidistant from vertices |
| Incenter | Angle bisectors | Equidistant from sides |
| Centroid | Medians | Balance point; 2:1 ratio |
| Orthocenter | Altitudes | Location varies by angle type |
Worked Examples
Orthocenter of a Right Triangle
Where is the orthocenter of a right triangle with the right angle at vertex ?
Two sides meeting at are already perpendicular. The altitudes from and to opposite sides pass through .
The orthocenter is at (the vertex of the right angle).
Coordinate Orthocenter
Find the orthocenter of , , .
Altitude from ⊥ : slope of → altitude slope = → line:
Altitude from ⊥ : slope of → altitude slope = → line:
Set equal: → →
Orthocenter =
Classifying the Triangle
The orthocenter of is outside the triangle. What can you conclude about ?
Orthocenter outside ↔ triangle is obtuse.
is an obtuse triangle.
Common Pitfalls
Confusing Altitude and Median
An altitude is perpendicular to the opposite side. A median hits the midpoint. They are NOT the same unless the triangle is isosceles (for the vertex between equal sides) or equilateral.
Altitude Foot Outside the Triangle
In an obtuse triangle, the foot of an altitude can fall outside the triangle (on the extension of the opposite side). Don't assume it always lands on the side.
Real-Life Applications
Construction — Finding Elevation
In roof construction, an altitude from the peak to the base gives the exact height of the roof. This measurement is essential for calculating attic space, ventilation, and snow load capacity.
Area Calculations
Area = ½ × base × height. The “height” is an altitude! You can use any side as the base with its corresponding altitude to compute the triangle's area.
Practice Quiz
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