Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
Pythagorean theorem used as a triangle "C" on its hypotenuse, and two similar right triangles ("A" and "B" ) constructed on the other two sides, formed by dividing the central triangle by its altitude. The sum of the areas of the two smaller triangles therefore is that of the third, thus "A" + "B" = "C" and reversing the above logic leads to the Pythagorean theorem a + b = c. ("See also Einstein's proof by dissection without rearrangement") The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of