Hypotenuse . It is the longest side of a right triangle , and the side opposite the right angle. The measure of the hypotenuse can be found using the Pythagorean theorem , if the length of the other two sides, called legs, is known .
Summary
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- 1 Etymology
- 2 Properties of the hypotenuse
- 3 orthogonal projections
- 4 Other properties
- 1 In whole numbers
- 5 Sources
Etymology
The word hypotenuse comes from the Greek term ὑποτείνουσα, a combination of hiccup “below” and teinein “lengthen.” Other authors suggest that the original meaning in Greek was due to an object that supports something, or the combination of hiccups “below” and tenuse “side.”
Properties of the hypotenuse
Pythagorean Theorem : It establishes that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs .
Orthogonal projections
- The lengthof the hypotenuse is equal to the sum of the lengths of the orthogonal projections of both legs.
- The square of the lengthof a leg is equal to the product of the length of its orthogonal projection on the hypotenuse times its length.
b² = a · mc² = a · n
- Also, the length of a legb is a proportional mean between the lengths of its projection m and that of the hypotenuse a.
a / b = b / ma / c = c / n
Other properties
- If a right triangle is inscribed in a circle, the hypotenuse coincides with the diameter and the midpoint of the hypotenuse is also the center of the circumscribed circle.
- The hypotenuse is used to define the sine, cosine, secant, and cosecant of an acute angle.
- If we circumscribe a right triangle of legs b and c, hypotenuse a in a circle of radius r, the equation a = b + c-2r fits.
- The area of a right triangle is A = = 0.5 ah, where a = hypotenuse and h = height.
- The hypotenuse is the axis of reflection of a right triangle.
- A diagonal of a rectangle divides it into two equal right triangles (shape and size), the common hypotenuse of which is the dividing diagonal.
In whole numbers
1.- The formula (k 2 – l 2 ) 2 +4 (kl) 2 = (k 2 + l 2 ) 2 , with integers k ≠ l, provides a = k 2 + l 2 , the hypotenuse, being k 2 – l 2 and 2kl the legs.
Example k = 7 and l = 3 gives: b = 40, c = 42 and a = 58.
2.- Another formula is doing l = 1, in the previous formula.
