Altitude (triangle)
[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 and perpendicular to a line containing the side or edge opposite the apex. This edge and line extension are called, respectively, the base and extended base of the altitude. The 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, 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 equals the triangle's area: /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, 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 divides the hypotenuse into two segments of lengths and. If we denote the length of the altitude by, we then have the relation
For acute triangles, the feet of the altitudes all fall on the triangle's sides. In an obtuse triangle, 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.
Theorems
The geometric altitude figures prominently in many important theorems and their proofs. For example, besides those theorems listed below, the altitude plays a central role in proofs of both the Law of sines and Law of cosines.Altitude in terms of the sides
For any triangle with sides and semiperimeter the altitude from side is given byThis follows from combining Heron's formula for the area of a triangle in terms of the sides with the area formula where the base is taken as side and the height is the altitude from the vertex .
By exchanging the on the denominator with or, this equation can also used to find the altitudes and, respectively.
Any two altitudes of a triangle are inversely proportional with the sides on which they fall.
Inradius theorems
Consider an arbitrary triangle with sides and with correspondingaltitudes. The altitudes and the incircle radius are related by
Circumradius theorem
Denoting the altitude from one side of a triangle as, the other two sides as and, and the triangle's circumradius as, the altitude is given byInterior point
If are the perpendicular distances from any point to the sides, and are the altitudes to the respective sides, thenArea theorem
Denoting the altitudes of any triangle from sides respectively as, and the semi-sum of the reciprocals of the altitudes as then the reciprocal of area isGeneral point on an altitude
If is any point on an altitude of any triangle, thenTriangle inequality
Since the area of the triangle is, the triangle inequality impliesSpecial cases
Equilateral triangle
From any point within an equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle. This is Viviani's theorem.Right triangle
In a right triangle with legs and and hypotenuse, each of the legs is also an altitude: and. The third altitude can be found by the relationThis is also known as the inverse Pythagorean theorem.
Note in particular: