Finding the Hypotenuse in Right Triangles

Finding the Hypotenuse in Right Triangles Warm-Up ? Lesson Question How can you find the length of the hypotenuse of a right triangle? Lesson Goals Find the hypotenuse of a right triangle. Use the Pythagorean theorem to solve for the unknown value. W 2K Evaluate square roots of perfect Recognize that nonperfect square roots can be approximated. squares. Words to Know Write the letter of the definition next to the matching word as you work through the lesson. You may use the glossary to help you. C leg A. the side of a right triangle that is opposite the right angle; always the longest side D right triangle B. a number that is the result of squaring a natural, or whole, number A hypotenuse C. in a right triangle, either of the two sides forming the right angle B perfect square D. a triangle having an interior angle measuring 90 degrees 1 Warm-Up Finding the Hypotenuse in Right Triangles Perfect Square Numbers • A number that can be produced by multiplying an integer by is known as a perfect square. itself • For example, the number 9 can be produced by multiplying the positive integer 3 by itself: (3 × 3) = 9. • The number 12, however, cannot be produced this way, so it is not a perfect square. Are the following numbers perfect squares? • 4 = (2 × 2) = • 25 = ( 5 22 × )=5 5 2 • 50 = not a perfect square 2 Instruction Finding the Hypotenuse in Right Triangles Slide 2 Longest Side of a Right Triangle A right triangle is a triangle that contains one right angle. The sides that make up the right angle are the legs, and the side the hypotenuse. The hypotenuse is always the triangle. opposite longest the right angle is side of a right c2 = a2 + b2 A A hypotenuse leg C leg b B C 3 c a B Instruction Finding the Hypotenuse in Right Triangles Slide 2 Using the Pythagorean Theorem to Find the Hypotenuse EXAMPLE Use the Pythagorean theorem to find the length of the hypotenuse, c. A 24 ft Pythagorean theorem: a2 + b2 = c2 49 c 72 + 242 = c2 + 576 = c 625 C 7 ft 2 = c2 625 = B ± 25 = c c= Distance is always positive. 4 c2 25 ft Instruction Finding the Hypotenuse in Right Triangles Slide 4 The Pythagorean Theorem and Distance REAL-WORLD CONNECTION To get to the local coffee shop, Van left his house, walked 3 blocks south, and then walked 4 blocks west. Van’s House 3 blocks south Coffee Shop 4 blocks west When he got to the coffee shop, Van realized he could take a more direct route home. How many blocks will Van have to walk to get home from the coffee shop if he takes the most direct route? a2 + b2 = c 2 9 32 + 42 = c2 + 16 = c 25 2 = c2 ± 25 = c 5 =c 5 blocks 5 Instruction Finding the Hypotenuse in Right Triangles Slide 7 Finding the Length of a Non-Perfect Hypotenuse EXAMPLE Use the Pythagorean theorem to find the length of the hypotenuse, c. A c= 2 in. a2 + b 2 = c 2 32 + ⎯ √ 13 22 = c2 + 4 =c 9 13 C B 3 in. 2 = c2 ± 13 = c 13 = c 10 Finding the Length of a Ladder REAL-WORLD CONNECTION Robert wants to hang a picture above his entertainment center. How long must a ladder be to reach a height of 13 feet if the bottom of the ladder is placed 3 feet from the base of the wall? a2 + b2 = c2 9+ c 32 + 132 = c2 169 = c2 178 = 13 ft 3 ft c2 178 = c Use a calculator to give an approximate number for the length. c≈ 6 13.3 ft Instruction Finding the Hypotenuse in Right Triangles Slide 12 Right Triangles and Measures REAL-WORLD CONNECTION Jenna is shopping for a new computer monitor. If she knows the measurements of the sides of the monitor, how can she figure out the diagonal measurement? a2 + b 2 = c 2 11 in. 112 c 121 + 14 in. + 142 = c 196 2 = c2 317 = c2 317 = The diagonal split the rectangle into 2 right triangles. 317 = c Pythagorean c≈ Use the theorem. 7 c2 17.8 in. Summary ? Lesson Question Finding the Hypotenuse in Right Triangles How can you find the length of the hypotenuse of a right triangle? Answer Use the Pythagorean theorem to write an equation substituting the value of the legs for a and b, and then solve for the length of the hypotenuse, c. Use this space to write any questions or thoughts about this lesson. 8