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Dissecting the right triangle along its altitude h yields two similar triangles, which can be augmented and arranged in two alternative ways into a larger right triangle with perpendicular sides of lengths p + h and q + h. One such arrangement requires a square of area h 2 to complete it, the other a rectangle of area pq. Since both ...
In a right triangle, the altitude drawn to the hypotenuse c divides the hypotenuse into two segments of lengths p and q. If we denote the length of the altitude by h c , we then have the relation h c = p q {\displaystyle h_{c}={\sqrt {pq}}} ( Geometric mean theorem ; see Special Cases , inverse Pythagorean theorem )
In the case of a right triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude. This property is known as the geometric mean theorem.
In a right triangle, the length of a cathetus is the geometric mean of the length of the adjacent segment cut by the altitude to the hypotenuse and the length of the whole hypotenuse. By the Pythagorean theorem, the sum of the squares of the lengths of the catheti is equal to the square of the length of the hypotenuse.
A right triangle ABC with its right angle at C, hypotenuse c, and legs a and b,. A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1 ⁄ 4 turn or 90 degrees).
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. [23] The length of the altitude is the distance between the base and the vertex.
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