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Images altitude geometry4/16/2023 ![]() ![]() does not have an angle greater than or equal to a right angle). The orthocenter lies inside the triangle if and only if the triangle is acute (i.e. The three (possibly extended) altitudes intersect in a single point, called the orthocenter of the triangle, usually denoted by H. Three altitudes intersecting at the orthocenter. 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. 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. If we denote the length of the altitude by hc, we then have the relation For acute and right triangles the feet of the altitudes all fall on the triangle's sides (not extended). In a right triangle, the altitude drawn to the hypotenuse c divides the hypotenuse into two segments of lengths p and q. It is common to mark the altitude with the letter h (as in height), often subscripted with the name of the side the altitude is drawn to. Also the altitude having the incongruent side as its base will be the angle bisector of the vertex angle. In an isosceles triangle (a triangle with two congruent sides), the altitude having the incongruent side as its base will have the midpoint of that side as its foot. The altitudes are also related to the sides of the triangle through the trigonometric functions. Thus, the longest altitude is perpendicular to the shortest side of the triangle. 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. It is a special case of orthogonal projection. The process of drawing the altitude from the vertex to the foot is known as dropping the altitude at that vertex. ![]() The length of the altitude, often simply called "the altitude", is the distance between the extended base and the vertex. ![]() The intersection of the extended base and the altitude is called the foot of the altitude. This line containing the opposite side is called the extended base of the altitude. In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). ![]()
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