Altitude (triangle)

In geometry, an altitude of a triangle is a line segment through a given vertex (called apex) and perpendicular to a line containing the side or edge opposite the apex.

This (finite) edge and (infinite) line extension are called, respectively, the base and extended base of the altitude.

The length of the altitude, often simply called "the altitude" or "height", symbol h, is the distance between the foot and the apex.

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 b) equals the triangle's area: A=hb/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.

If we denote the length of the altitude by hc, we then have the relation 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.

the altitude from side a (the base) is given by This follows from combining Heron's formula for the area of a triangle in terms of the sides with the area formula

By exchanging a with b or c, this equation can also used to find the altitudes hb and hc, respectively.

Consider an arbitrary triangle with sides a, b, c and with corresponding altitudes ha, hb, hc.

The altitudes and the incircle radius r are related by[3]: Lemma 1 Denoting the altitude from one side of a triangle as ha, the other two sides as b and c, and the triangle's circumradius (radius of the triangle's circumscribed circle) as R, the altitude is given by[4] If p1, p2, p3 are the perpendicular distances from any point P to the sides, and h1, h2, h3 are the altitudes to the respective sides, then[5] Denoting the altitudes of any triangle from sides a, b, c respectively as ha, hb, hc, and denoting the semi-sum of the reciprocals of the altitudes as

implies[8] From any point P within an equilateral triangle, the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.

The third altitude can be found by the relation[9][10] This is also known as the inverse Pythagorean theorem.

The altitude from A (dashed line segment) intersects the extended base at D (a point outside the triangle).
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.
The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle.
The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides ( p + q , r , s ) , ( r , p , h ) and ( s , h , q ) ,
Comparison of the inverse Pythagorean theorem with the Pythagorean theorem