![]() The altitudes from each of the acute angles of an obtuse triangle lie entirely outside the triangle, as does the orthocenter H.įor acute triangles the feet of the altitudes all fall on the triangle's sides (not extended). ( p + q ) 2 = r 2 + s 2 p 2 + 2 p q + q 2 = p 2 + h 2 ⏞ + h 2 + q 2 ⏞ 2 p q = 2 h 2 ∴ h = p q ( Geometric mean theorem) 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. ![]() ![]() Using Pythagoras' theorem on the 3 triangles of sides ( p + q, r, s ), ( r, p, h ) and ( s, h, q ), 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. 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. Perpendicular line segment from a triangle's side to opposite vertex The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle.Īltitudes 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. ![]()
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