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Right Triangle

A right triangle is a triangle with an angle of ( radians). The sides , , and of such a triangle satisfy the Pythagorean theorem

(1)

where the largest side is conventionally denoted and is called the hypotenuse. The other two sides of lengths and are called legs, or sometimes catheti.

The right angle is conventionally indicated in a diagram by a small square called a right-angle marker (Marecek et al. 2020, §9.3).

The favorite A-level math exam question of the protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time asks for proof that a triangle with sides of the form , , and where is a right triangle, and that the converse does not hold (Haddon 2003, pp. 214 and 223-226).

The side lengths of a right triangle form a Pythagorean triple when all three are integers. A triangle that is not a right triangle is sometimes called an oblique triangle. Special cases of the right triangle include the isosceles right triangle (middle figure) and 30-60-90 triangle (right figure).

For any three similar shapes of area on the sides of a right triangle,

(2)

which is equivalent to the Pythagorean theorem.

For a right triangle with sides , , and hypotenuse , the area is simply

(3)

The inradius can be found by equating the area of the triangle with the sum of the areas of the three triangles , , and having the inradii as altitudes, giving

(4)

Solving for then gives

(5)

This can also be written in the equivalent forms

			(6)
			(7)

The hypotenuse of a right triangle is a diameter of the triangle's circumcircle, so the circumradius is given by

(8)

Its outer Soddy circle has geometric radius equal to the semiperimeter (F. M. Jackson, pers. comm., Sep. 4, 2010).

A primitive right triangle is a right triangle having integer sides , , and such that , where is the greatest common divisor. The set of values is then known as a primitive Pythagorean triple.

For a right triangle with integer side lengths, any primitive Pythagorean triple can be written

			(9)
			(10)
			(11)

Using these, equation (6) becomes

			(12)
			(13)

which is an integer whenever and are integers (Ogilvy and Anderson 1988, p. 68).

Given a right triangle , draw the altitude from the right angle . Then the triangles and are similar.

In a right triangle, the midpoint of the hypotenuse is equidistant from the three polygon vertices (Dunham 1990). A short proof follows from the fact that the hypotenuse is a diameter of the circumcircle. Its midpoint is therefore the circumcenter, so its distances to the three vertices equal the circumradius.

An elementary proof is as follows. Given , let be the midpoint of (so that ). Draw , then since is similar to , it follows that . Since both and are right triangles and the corresponding legs are equal, the hypotenuses are also equal, so we have and the theorem is proved.

In addition, the triangle median and altitude of a triangle are reflections about the angle bisector of iff is a right triangle (G. McRae, pers. comm., May 1, 2006).

Fermat showed how to construct an arbitrary number of equiareal nonprimitive right triangles. An analysis of Pythagorean triples demonstrates that the right triangle generated by a Pythagorean triple has common area

(14)

(Beiler 1966, pp. 126-127). The only extremum of this function occurs at . Since for , the smallest area shared by three nonprimitive right triangles is given by , which results in an area of 840 and corresponds to the triplets (24, 70, 74), (40, 42, 58), and (15, 112, 113) (Beiler 1966, p. 126). There are also quartets of right triangles with the same area. The quartet having the smallest known area is (111, 6160, 6161), (231, 2960, 2969), (518, 1320, 1418), (280, 2442, 2458), with area (Beiler 1966, p. 127). Guy (1994) gives additional information.

It is also possible to find sets of three and four right triangles having the same perimeter (Beiler 1966, pp. 131-132).

In a given right triangle, an infinite sequence of squares that alternately lie on the hypotenuse and longest leg can be constructed, as illustrated above. These create a sequence of increasingly smaller similar right triangles. Let the original triangle have legs of lengths and and hypotenuse of length . Also define

			(15)
			(16)

Then the sides of the th square are of length

(17)

Number the upper left triangle as 1, and then the remainder by following the "strip" of triangles at adjoining vertices. Then the side lengths of these triangles are

		${s_((n+1)/2) for n odd; (ab)/cx^(n/2) for n even$	(18)
		${(b^2)/ax^((n+1)/2) for n odd; (b^2)/cx^(n/2) for n even$	(19)
		${(bc)/ax^((n+1)/2) for n odd; s_(n/2) for n even.$	(20)

The inradii of the corresponding triangles can be found from

(21)

giving

$r_n={b/ayx^((n+1)/2) for n odd; b/cyx^(n/2) for n even.$

(22)

A Sangaku problem from 1913 in the Miyagi Prefecture asks for the relationships between the first, third, and fifth inradii (Rothman 1998). This can be solved using elementary trigonometry as well as the explicit equations given above, and has solution

(23)

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References

Beiler, A. H. "The Eternal Triangle." Ch. 14 in Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. New York: Dover, 1966.Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987.Dunham, W. Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120-121, 1990.Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, pp. 160-161, 1984.Guy, R. K. "Triangles with Integer Sides, Medians, and Area." §D21 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 188-190, 1994.Haddon, M. The Curious Incident of the Dog in the Night-Time. New York: Vintage, 2003.Kern, W. F. and Bland, J. R. Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 2, 1948.Marecek, L.; Anthony-Smith, M.; and Mathis, A. H. "Use Properties of Angles, Triangles, and the Pythagorean Theorem." §9.3 in Prealgebra 2e. Houston, TX: OpenStax, 2020. https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem.Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, p. 68, 1988.Rothman, T. "Japanese Temple Geometry." Sci. Amer. 278, 85-91, May 1998.Sierpiński, W. Pythagorean Triangles. New York: Academic Press, 1962.Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 155-161, 1943a.Whitlock, W. P. Jr. "Rational Right Triangles with Equal Areas." Scripta Math. 9, 265-268, 1943b.

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Right Triangle

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Weisstein, Eric W. "Right Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RightTriangle.html

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Right Triangle

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