Exploring the Pythagorean Theorem Warm up Answer Key

Warm-Up ? Exploring the Pythagorean Theorem Lesson Question What are properties of right triangles? Lesson Goals Identify and examine right triangles. Recognize perfect squares . Identify sets of Pythagorean triples W 2K Apply the Pythagorean theorem. . Words to Know Fill in this table as you work through the lesson. You may also use the glossary to help you. perfect square a number that is the result of squaring a natural, or whole, number Pythagorean triples a set of three positive integers that satisfy the Pythagorean theorem and are possible side lengths in a right triangle Pythagorean theorem the theorem stating that the sum of the squares of the lengths of the legs in a right triangle is equal to the square of the length of the hypotenuse hypotenuse right triangle legs the side of a right triangle that is opposite the right angle; always the longest side a triangle having an interior angle measuring 90 degrees in a right triangle, either of the two sides forming the right angle 1 Warm-Up Exploring the Pythagorean Theorem Parts of a Right Triangle A right triangle is a triangle that contains one right angle, which measures 90 degrees. The sides that make up the right angle are the legs, and the side opposite the right angle is the hypotenuse. Right triangle ABC A leg hypotenuse C • Right triangles can be isosceles triangles where we have one right angle and the other two angles are each degrees. 45 B leg • It is possible that a right • Right triangles triangle is a scalene triangle. In this case, all the cannot sides are lengths. • A right triangle cannot be obtuse 2 different . be equilateral. • A right triangle cannot be acute. Instruction Exploring the Pythagorean Theorem Slide 2 Perfect Squares squaring A perfect square is a number that is the result of whole number. a natural, or 4 4 6 16 4 ⋅ 4 = 42 = 6 6 ⋅ 6 = 62 = 36 Perfect Squares Versus Not Perfect Squares EXAMPLE • Perfect Squares • 4 = 22 • 25 = • Not Perfect Squares • 18 52 • 42 • 64 = 82 • 3 42 < 18 < 52 between 62 and 80 72 between 82 and 92 Instruction Exploring the Pythagorean Theorem Slide 5 Perfect Squares of Right Triangles EXAMPLE Squares right can be created from the side lengths of a triangle. 3 3 5 3 32 = 9 4 9 + 16 = 25 4 4 16 42 = 4 Instruction Exploring the Pythagorean Theorem Slide 5 Pythagorean theorem For any right triangle with legs a and b and a2 + b2 hypotenuse c, = c2 52 32 + 4 2 = 9 + 16 = 25 25 = 25 3 25 5 9 5 4 16 9 + 16 = 25 5 5 Instruction Exploring the Pythagorean Theorem Slide 8 Sides of a Particular Right Triangle Examine right triangle ABC with side lengths 8, 15, and 17. A b = 15 c = 17 a=8 C B The diagram shows: + 152 = 17 82 • 2 • 64 + 225 = 289 • a2 + b2 = c2 When three whole numbers form the sides of a right triangle, they are called Pythagorean triples. So 8, 15 and 17 represent a Pythagorean 6 triple . Instruction Exploring the Pythagorean Theorem Slide 8 Pythagorean Triples A Pythagorean triple is a set of three Pythagorean theorem and are possible • Side lengths 3, 4, and 5 32 + 42 = 52. 122 = 132 25 + 144 = 169 • (7, 24, 25) 169 = 169ü 72 + 242 = integers that satisfy the side lengths of a right triangle. is a Pythagorean triple because We can also use whole number multiples of known Pythagorean triples to develop other Pythagorean triples. If I multiply the triple (3, 4, 5) by 3, I get: • (5, 12, 13) 52 + positive ( 9 92 + 252 49 + 576 = 625 625 = 625ü , 12, 15) 122 = 152 81 + 144 = 225 225 = 225ü Since I can choose any whole number to multiply by in creating these triples, I know that there are infinitely many Pythagorean triples. 7 Instruction Exploring the Pythagorean Theorem Slide 11 Real-World Use of the Pythagorean Theorem Stella purchased a small plot of land with dimensions of 30 yards, 16 yards, and 34 yards. How can we determine if this triangular lot forms a right triangle? a2 30 yards 34 yards 900 + 16 yards + b2 = c 302 + 162 = 256 2 342 = 1156 1156 = 1156 These side lengths satisfy the Pythagorean theorem. Therefore, her plot of land is in the shape of a right triangle. 8 Instruction Exploring the Pythagorean Theorem Slide 13 Real-World Use of Pythagorean Triples Carson found a piece of wood he wants to use to make a table top. The wood measures 40 inches by 10 inches. The diagonal of the wood is 41 inches. Is the piece of wood rectangular? a2 + b2 = c2 ? 102 + 402 = 100 + 41” 10” 40” 412 1600 = 1681 1700 ≠ 1681 ? If this is a right triangle, then 10, 40, and 41 must satisfy the Pythagorean theorem. So, the piece of wood he has is a non-rectangular parallelogram . The numbers 10, 40, and 41 are close to the Pythagorean triple 9, 40, and 41. 9 Summary ? Lesson Question Exploring the Pythagorean Theorem What are properties of right triangles? Answer For triangles with one right angle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. The Pythagorean theorem proves this. Use this space to write any questions or thoughts about this lesson. 10