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THE PYTHAGOREAN THEOREM What Is the Pythagorean Theorem? The Pythagorean Theorem is a theorem that states the sum of the squares of the legs of a right triangle equals the square of the length of the hypotenuse. • In mathematical terms, this is represented by a² + b² = c², where a and b are the length of the legs and c is the length of the hypotenuse of a right triangle. • A Pythagorean triple is a set of numbers that always satisfy the equation, a² + b² = c². • By using the Pythagorean Theorem, a triangle can be determined to be a right triangle. Also if either the legs or a leg and hypotenuse of a right triangle are given, the Pythagorean theorem can be used to find the missing leg or hypotenuse. In order to use the Pythagorean Theorem, it is important to know how to evaluate a power. A power is a number in the form of x², which means x squared or x · x. With the Pythagorean Theorem, a number will be equal to c², for example, and need to be solved for c. In order to solve for c, the square root must be taken. The square root of a number is equal to a number that, when multiplied by itself, will produce the given number. The Pythagorean Theorem can be applied to any right triangle. Many everyday items can be evaluated using the Pythagorean Theorem. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.

How to use the Pythagorean Theorem In order to use the Pythagorean Theorem, powers and roots should be looked at first. For example, what is 7² and the square root of 225? Ex. 7² = 7 · 7 = 49 The square root of 225 = √225 = 15 because 15 · 15 = 225. The Pythagorean Theorem can only be used if a triangle is a right triangle. This would have to be stated in order to use the Pythagorean Theorem. If a triangle is a right triangle and the sides are given, the hypotenuse can be found as follows: Ex. A right triangle has sides of 6 cm and 8 cm. a² + b² = c² 6² + 8² = c² 36 + 64 = c² 100 = c² √100 = c² ; c =10 The hypotenuse of the triangle is 10 cm. The Pythagorean Theorem can also be used if one leg and the hypotenuse are given to get the missing leg. The Pythagorean Theorem is also used to prove that triangles are right triangles. • For example if a triangle has legs that are 5 cm and 10 cm and a hypotenuse of 15, is the triangle a right triangle? Ex. a² + b² = c² 5² + 10² = 15² 25 + 100 = 225 125 ≠ 225 so the triangle is not a right triangle. © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.

Objects in every day life can be used with the Pythagorean Theorem also. • For example, if an 18 ft ladder leans against a house and rests on the house16 ft above the ground, how far away is the ladder from the house given that the house and the ground make a right angle? Ex. a² + b² = c² a² + 16² = 18² a² + 256 = 324 a² = 68 a = √68 ≈ 8.25 ≈ 8.3 ft. The ladder is approximately 8.3 ft away from the house. Try This! 1. What is 8³? 2. What is the square root of 3600? 3. A triangle is a right triangle. The legs are 8 cm and 14 cm. What is the length of the hypotenuse? 4. A right triangle has a hypotenuse of 26 ft and a leg that is 24 ft. What is the length of the missing leg? 5. A triangle has sides of 5 cm and 8 cm. The hypotenuse is 13. Is the triangle a right triangle? 6. A square is cut into two triangles by a diagonal. If a side of the square measures 10 m, what is the length of the diagonal? © Copyright NewPath Learning. All Rights Reserved. Permission is granted for the purchaser to print copies for non-commercial educational purposes only. Visit us at www.NewPathLearning.com.