In a right triangle, the hypotenuse is the side of the triangle which is opposite the right angle. It is also the longest side of the triangle. The other two sides are sometimes called the "legs" of the right triangle. The Greek word hupoteinousa, from which we derive "hypotenuse," means, roughly, a stretched cord, invoking the image of a rope stretched tightly between two stakes in the ground. The hypotenuse plays an essential role in what is perhaps the most famous of all mathematical theorems, the Pythagorean Theorem. This theorem, attributed to the Greek philosopher and mathematician, Pythagoras (c 569-c 475 BC), states that if the legs of a right triangle are denoted by a and b and the hypotenuse by c, then c2=a2+b2, or, in English, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse became a source of great consternation to Pythagoras, who was something of a mystic in his study of numbers. Pythagoras and his followers believed that numbers ruled the universe, and by "numbers" they meant the natural numbers 1,2,3, etc. For certain right triangles, the hypotenuse was indeed one of these natural numbers. For example, a right triangle with legs of lengths 3 units and 4 units has a hypotenuse of length 5 units, since 32+42=52. For this reason the numbers 3, 4, and 5 are collectively called a Pythagorean triple. Other examples include 5, 12, and 13; 8, 15, and 17; and 7, 24, and 25. In fact, there are an infinite number of Pythagorean triples or, to put it another way, there are an infinite number of right triangles with the lengths of all three sides being natural numbers. These triangles were very pleasing to Pythagoras as they fit in very nicely with his theories about the world being ruled by natural numbers.
Unfortunately, for Pythagoras, there are also an infinte number of right triangles which do not have this property. In particular, Pythagoras noticed that a right triangle whose legs are each 1 unit long will have a hypotenuse with length 2, which is not a natural number. Furthermore, 2 is a non-repeating infinite decimal and cannot be expressed as the ratio of two natural numbers. For this reason, Pythagoras called such numbers incommensurable, and he was evidently quite troubled by their appearance as lengths. How could such a number represent a length? Where exactly does one find such a number on the real number line? The incommensurables, which we now call irrational numbers, did not fit nicely with Pythagoras' doctrine that all quantities should be natural numbers or ratios of natural numbers. Legend says that Pythagoras instructed his followers not to discuss the incommensurability of the hypotenuse with the legs in certain right triangles. As far as we know, Pythagoras never tried to develop a theory which would bring the irrational numbers into his view that "number" was the ruling concept of the universe.
The hypotenuse is also involved in the definition of the so-called trigonometric ratios for right triangles. The sine of an angle is defined to be the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The tangent ratio is then defined to be the ratio of the sine to the cosine. These ratios combined with the Pythagorean theorem may be used to determine the lengths of sides or the measure of angles in a right triangle when other sides and angles are known. For example, if we know that the hypotenuse of a right triangle is 20 meters long and that one angle of the triangle has a measure of 30°, then we can determine the length of the adjacent side, a, to the angle using the equation a/20=sin(30°), from which it follows that a=20sin(30°)=10 meters. Similarly, we could determine the length of the opposite side from the angle by using the cosine ratio. Then, as a check of our computations, we could add the square of the adjacent side's length to the square of the opposite side's length which should give us 400 or the square of the length of the hypotenuse.
We might also note that the familiar "distance formula" for finding the distance between two points in the plane is nothing more than a computation of the length of the hypotenuse of the right triangle whose legs are formed by drawing perpendicular segments from the two points to the intersection of the two segments. Thus, the distance between the points (2,3) and (4,7), by the distance formula is ((4-2)2+(7-3)2)=20; but if we constuct the right triangle with vertices (2,3), (4,7), and (4,3), we see that its legs have lengths of 2 and 4 units, so that the Pythagorean Theorem gives (hypotenuse)2=22+42=4+16=20. Therefore, the hypotenuse has length 20, which agrees with the distance formula computaion.
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