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The side lengths of a Pythagorean triple are integers that fit the Pythagorean Theorem. These special triangles appear frequently in geometry text books and on standardized tests like the SAT and the GRE.  If you memorize the first 2 Pythagorean triples, in particular, you can save yourself a lot of time on these tests because you can immediately know the hypotenuse of one of these triangles just by looking at the side lengths!   The first Pythagorean triple is 3-4-5 (32 + 42 = 52, 9 + 16 = 25).  When you see a right triangle with legs of length 3 and 4, you can instantly be certain that the hypotenuse will be 5 without having to do any calculations. The ratio of a Pythagorean triple holds true even when the sides are multiplied by another number.  For example a right triangle with legs of length 6 and 8 will have a hypotenuse of 10 (62 + 82 = 102, 36 + 64 = 100).  The same holds true for 9-12-15, and even 1.5-2-2.5.  Try the math and see for yourself! The second Pythagorean triple that commonly appears on tests is 5-12-13 (52 + 122 = 132, 25 + 144 = 169).  Also be on the lookout for multiples like 10-24-26 and 2.5-6-6.5. A 45-45-90 right triangle has angles of 45, 45, and 90 degrees, and is also called an Isosceles Right Triangle.  It occurs frequently on standardized tests, and is a very easy triangle to solve.  The ratio between the sides of this triangle is 1:1:Sqrt(2), which means that the length of the legs are equal, and the length of the hypotenuse is simply the leg length multiplied by the square root of two.  To calculate the hypotenuse of this triangle based on the length of one of the legs, simply multiply the leg length by Sqrt(2). Knowing this ratio comes in especially handy when your test or homework question gives you the side lengths in terms of variables instead of integers. This triangle has angle measurements of 30, 60, and 90 degrees, and occurs when you cut an equilateral triangle in half.  The sides of the 30-60-90 right triangle always maintain the ratio 1:Sqrt(3):2, or x:Sqrt(3)x:2x.  If you are given the length of one leg of 30-60-90 right triangle and are asked to find the hypotenuse, it is very easy to do:  If you are given the length of the shortest leg (opposite the 30-degree angle,) simply multiply the leg length by 2 to find the length of the hypotenuse.  For instance, if the length of the shortest leg is 4, you know that the hypotenuse length must be 8. If you are given the length of the longer leg (opposite the 60-degree angle,) multiply that length by 2/Sqrt(3) to find the length of the hypotenuse.  For instance, if the length of the longer leg is 4, you know that the hypotenuse length must be 4.62.
Learn to recognize Pythagorean Triple Triangles. Memorize the side ratios of a 45-45-90 right triangle. Learn the side ratios of a 30-60-90 right triangle.