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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 b) equals the triangle's area: A = h b /2 ...
The only triangle with consecutive integers for an altitude and the sides is acute, having sides (13,14,15) and altitude from side 14 equal to 12. The smallest-perimeter triangle with integer sides in arithmetic progression, and the smallest-perimeter integer-sided triangle with distinct sides, is obtuse: namely the one with sides (2, 3, 4).
The extended base of a triangle (a particular case of an extended side) is the line that contains the base. When the triangle is obtuse and the base is chosen to be one of the sides adjacent to the obtuse angle, then the altitude dropped perpendicularly from the apex to the base intersects the extended base outside of the triangle.
The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when =, = /, and the altitude of the triangle from the base of length is equal to . The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2 2 / 3 {\displaystyle 2 ...
If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal, then the triangles are congruent (side-angle-side). The area of a triangle is half the area of any parallelogram on the same base and having the same altitude.
Let D be a point on the line BC, not equal to B or C and such that AD is not an altitude of triangle ABC. Let B 1 be the base (foot) of the altitude in the triangle ABD through B and let C 1 be the base of the altitude in the triangle ACD through C. Then, if D is strictly between B and C, one and only one of B 1 or C 1 lies inside ABC and it ...
The SEC didn't get four teams in the field, but the Longhorns only lost to one team this year (Georgia), and they've got a very reasonable path to the semifinals as the highest-ranked at-large team.
Points J, K, L are the midpoints of the line segments between each altitude's vertex intersection (points A, B, C) and the triangle's orthocenter (point S). For an acute triangle , six of the points (the midpoints and altitude feet) lie on the triangle itself; for an obtuse triangle two of the altitudes have feet outside the triangle, but these ...