Editing Altitude (triangle) (section)

{{short description|Perpendicular line segment from a triangle's side to opposite vertex}}
[[Image:Projection formula (3).png|thumb|The altitude from A (dashed line segment) intersects the extended base at D (a point outside the triangle).]]

In [[geometry]], an '''altitude''' of a [[triangle]]  is a [[line segment]] through a given [[Vertex (geometry)|vertex]] (called ''[[apex (geometry)|apex]]'') and [[perpendicular]] to a [[line (geometry)|line]] containing the side or [[edge (geometry)|edge]] opposite the apex. This (finite) edge and (infinite) line extension are called, respectively, the ''[[base (geometry)|base]]'' and ''[[extended side|extended base]]'' of the altitude. The [[point (geometry)|point]] at the intersection of the extended base and the altitude is called the '''''foot''''' of the altitude. The length of the altitude, often simply called "the altitude" or "height", symbol {{mvar|h}}, is the distance between the foot and the apex.  The process of drawing the altitude from a vertex to the foot is known as ''dropping the altitude'' at that vertex. It is a special case of [[orthogonal projection]].

Altitudes can be used in the computation of the [[area of a triangle]]: one-half of the product of an altitude's length and its base's length (symbol {{mvar|b}}) equals the triangle's area: {{mvar|A}}{{=}}{{mvar|h}}{{mvar|b}}/2. Thus, the longest altitude is perpendicular to the shortest side of the triangle. The altitudes are also related to the sides of the triangle through the [[trigonometric functions]].

In an [[isosceles triangle]] (a triangle with two [[congruence (geometry)|congruent]] sides), the altitude having the incongruent side as its base will have the [[midpoint]] of that side as its foot. Also the altitude having the incongruent side as its base will be the [[angle bisector]] of the vertex angle.

In a [[right triangle]], the altitude drawn to the [[hypotenuse]] {{mvar|c}} divides the hypotenuse into two segments of lengths {{mvar|p}} and {{mvar|q}}. If we denote the length of the altitude by {{mvar|h{{sub|c}}}}, we then have the relation
:<math>h_c=\sqrt{pq} </math>&nbsp; ([[Geometric mean theorem]]; see [[#Right_triangle|Special Cases]], [[inverse Pythagorean theorem]])

[[File:Rtriangle.svg|thumb|right|In a right triangle, the altitude from each acute angle coincides with a leg and intersects the opposite side at (has its foot at) the right-angled vertex, which is the orthocenter.]]
For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an [[obtuse angle]]), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite [[extended side]], exterior to the triangle. This is illustrated in the adjacent diagram: in this obtuse triangle, an altitude dropped perpendicularly from the top vertex, which has an acute angle, intersects the extended horizontal side outside the triangle.