The Pythagorean TheoremDate: 07/07/97 at 13:57:09 From: Noelle Subject: Geometry I'm having problems with the Pythagorean Theorem! My teacher has explained it but I still don't get it. Some of it is a breeze but other parts just stump me! Thanks, Noelle Date: 09/19/97 at 13:15:00 From: Doctor Chita Subject: Re: Geometry Hi Noelle: More than 2,000 years ago Pythagoras came up with an extraordinary discovery about the relation among the squares on the sides of a right triangle. You can understand what he discovered by following the next four steps: 1. Draw a right triangle. The two sides that form the right angle are called "legs". The third side, opposite the right angle, is called the hypotenuse. 2. Label the vertices of the triangle A, B, and C, with C at the right angle. Then label the sides opposite each vertex a, b, and c. If you did this correctly, the two legs are a and b, and the hypotenuse is c. 3. Now draw a square on each side of the triangle. The areas of the squares are a^2, b^2, and c^2. (The symbol ^ means "raise to a power." In this case, square the numbers.) 4. What Pythagoras demonstrated was that the sum of the areas of the squares on the legs is equal to the area of the largest square. Using our labels, you can write: a^2 + b^2 = c^2. This is Pythagoras' theorem. It's important to remember that we don't have to use A, B, and C (and a, b, and c) as our labels. Any labels will do. In general the Theorem says that:leg(1)^2 + leg(2)^2 = hypotenuse^2. When using the theorem to solve problems it's important to locate the right angle in the triangle and then identify the legs and the hypotenuse. Once you have those, you can use the formula. You can solve many problems involving right triangles by knowing any two of the three sides. For example, suppose the sides you are given are the two legs, and they measure 3 and 4. Then the unknown side is the hypotenuse. Substitute 3 and 4 for a and b in the theorem and solve for c. 3^2 + 4^2 = c^2 9 + 16 = c^2 25 = c^2 5 = c (Actually, there are two answers to c^2 = 25. One is 5 and the other is -5. But in geometry we only use positive lengths so we can disregard -5.) If the two sides you are given are a leg and the hypotenuse, then to find the length of the unknown leg, you have an extra algebra step to complete. Let's say a = 5 and c = 13, where a is a leg and c is the hypotenuse. Then, a^2 + b^2 = c^2 5^2 + b^2 = 13^2 25 + b^2 = 169 Subtract 25 from both sides of the equation: b^2 = 169 - 25 = 144 b = 12. (Throwing away -12 as a length.) Does this help? -Doctor Chita, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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