Altitude

In geometry, the altitude of a geometric figure generally refers to a perpendicular distance measured from the base of the figure to its opposite side. The term altitude is often used interchangeably with "height." Below are a few examples depicting the altitudes of some geometric figures.

The dotted red lines in the figures above represent their altitudes. Note that the altitude can be depicted at multiple points within the figures, not just the ones specifically shown.

Altitude in triangles

Altitude in triangles is defined slightly differently than altitude in other geometric figures. In other geometric figures, such as those shown above (except for the cone), the altitude can be formed at multiple points in the figure. In a triangle however, the altitude must pass through one of its vertices, and the line segment connecting the vertex and the base must be perpendicular to the base. In other words, an altitude in a triangle is defined as the perpendicular distance from a base of a triangle to the vertex opposite the base. An altitude of the isosceles triangle is shown in the figure below:

Since all triangles have 3 vertices, every triangle has 3 altitudes, as shown in the figure below:

The base of a triangle is determined relative to a vertex of the triangle; the base is the side of the triangle opposite the chosen vertex. Each of the altitudes of a triangle forms a right triangle, and the altitudes of a triangle all intersect at a point referred to as the orthocenter. Along with the use of trigonometric relationships, the altitudes of a triangle can be used to determine many characteristics of triangles.

It is also worth noting that the position of the orthocenter changes depending on the type of triangle; for a right triangle, the orthocenter is at the vertex containing the right angle; for an obtuse triangle, the orthocenter is outside the triangle, opposite the longest side; for an acute triangle, the orthocenter is within the triangle.