Overview

Altitudes are perpendicular lines from vertices to opposite sides. The three altitudes meet at the orthocenter. Altitude lengths connect directly to area, so they are a fast way to solve for heights.

Key Ideas

• Area $A = \frac{1}{2}ah_a$, so $h_a = \frac{2A}{a}$.
• In an acute triangle, the orthocenter is inside the triangle.
• In a right triangle, the orthocenter is the right-angle vertex.
• In an obtuse triangle, the orthocenter lies outside the triangle.

Core Skills

Use Area to Find Heights

If you know the area and a base, compute the corresponding altitude with $h = 2A/b$.

Recognize Orthocenter Location

The triangle type determines where the orthocenter is: inside for acute, on a vertex for right, and outside for obtuse.

Combine with Similarity

Dropping an altitude often creates similar right triangles that unlock ratios and lengths.

Worked Example

Triangle $ABC$ has area $24$ and base $BC=8$. Find the altitude from $A$.

$24 = \frac{1}{2}\cdot 8 \cdot h$, so $h=6$.

More Examples

Example 1: Right Triangle Orthocenter

In a right triangle $ABC$ with right angle at $C$, where is the orthocenter?

The orthocenter is $C$.

Example 2: Area and Alternate Base

Triangle $ABC$ has area $30$ and side $AC=10$. Find the altitude from $B$.

$h_b = 2A/AC = 60/10 = 6$.

Strategy Checklist

• Mark which segment is the base for the altitude you need.
• Use $A=\tfrac{1}{2}bh$ before introducing extra variables.
• Decide triangle type to place the orthocenter correctly.
• If an altitude creates similar triangles, compare ratios first.

Common Pitfalls

• Forgetting the factor of $1/2$ in area.
• Confusing altitude with median.
• Using the wrong base for the altitude.

Practice Problems