9 8 The Pythagorean Theorem Warm Up Problem

9 -8 The Pythagorean Theorem Warm Up Problem of the Day Lesson Presentation Course 2

9 -8 The Pythagorean Theorem Warm Up Estimate each square root to the nearest whole number. Use a calculator to check the reasonableness of your answers. 1. 2. 3. 4. Course 2 √ 18 √ 26 √ 86 √ 125 4 5 9 11

9 -8 The Pythagorean Theorem Problem of the Day A shipping carton measures 12 in. by 15 in. by 16 in. What is the longest rod that can be shipped in it? 25 in. Course 2

9 -8 The Pythagorean Theorem Learn to use the Pythagorean Theorem to find the length of a side of a right triangle. Course 2

9 -8 The Insert Lesson Title Here Pythagorean Theorem Vocabulary leg hypotenuse Pythagorean Theorem Course 2

9 -8 The Pythagorean Theorem In a right triangle, the two sides that form the right angle are called legs. The side opposite the right angle is called the hypotenuse. Leg Hypotenuse Leg One of the first people to recognize the relationship between the sides of a right triangle was the Greek mathematician Pythagoras. This special relationship is called the Pythagorean Theorem. Course 2

9 -8 The Pythagorean Theorem PYTHAGOREAN THEOREM In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. a 2 + b 2 = c 2 c a b You can use the Pythagorean Theorem to find the length of any side of a right triangle. Course 2

9 -8 The Pythagorean Theorem Additional Example 1 A: Calculating the Length of a Side of a Right Triangle Use the Pythagorean Theorem to find the missing measure. 12 cm c 16 cm a 2 + b 2 = c 2 122 + 162 = c 2 144 + 256 = c 2 400 = c 2 √ 400 = √c 2 Use the Pythagorean Theorem. Substitute for a and b. Evaluate the powers. Add. Take the square root of both sides. 20 = c The length of the hypotenuse is 20 cm. Course 2

9 -8 The Pythagorean Theorem Additional Example 1 B: Calculating the Length of a Missing Side of a Right Triangle Use the Pythagorean Theorem to find the missing measure. b 13 cm a 2 + b 2 = c 2 52 + b 2 = 132 25 + b 2 = 169 – 25 b 2 = 144 5 cm Use the Pythagorean Theorem. Substitute for a and c. Evaluate the powers. Subtract 25 from each side. √b 2 = √ 144 Take the square root of both sides. b = 12 The length of the missing leg is 12 cm. Course 2

9 -8 The Pythagorean Theorem Check It Out: Example 1 A Use the Pythagorean Theorem to find the missing measure. 11 cm c 15 cm a 2 + b 2 = c 2 112 + 152 = c 2 121 + 225 = c 2 346 = c 2 √ 346 = √c 2 Use the Pythagorean Theorem. Substitute for a and b. Evaluate the powers. Add. Take the square root of both sides. 18. 6 c The length of the hypotenuse is about 18. 6 cm. Course 2

9 -8 The Pythagorean Theorem Check It Out: Example 1 B Use the Pythagorean Theorem to find the missing measure. b 5 cm a 2 + b 2 = c 2 3 2 + b 2 = 5 2 9 + b 2 = 25 – 9 b 2 = 16 3 cm Use the Pythagorean Theorem. Substitute for a and c. Evaluate the powers. Subtract 9 from each side. √b 2 = √ 16 Take the square root of both sides. b = 4 The length of the missing leg is 4 cm. Course 2

9 -8 The Pythagorean Theorem Additional Example 2: Problem Solving Application A square field has sides of 75 feet. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth. Course 2

9 -8 The Pythagorean Theorem Additional Example 2 Continued 1 Understand the Problem Rewrite the question as a statement. • Find the distance from one corner of the field to the opposite corner of the field. List the important information: • Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles. • The segment between the two corners is the hypotenuse. • The sides of the field are legs, and they are each 75 feet long. Course 2

9 -8 The Pythagorean Theorem Additional Example 2 Continued 2 Course 2 Make a Plan You can use the Pythagorean Theorem to write an equation.

9 -8 The Pythagorean Theorem Additional Example 2 Continued 3 Solve a 2 + b 2 = c 2 Use the Pythagorean Theorem. 752 + 752 = c 2 Substitute for the known variables. 5, 625 + 5, 625 = c 2 Evaluate the powers. 11, 250 = c 2 Add. 106. 066012 c 106. 1 c Take the square roots of both sides. Round. The distance from one corner of the field to the opposite corner is about 106. 1 feet Course 2

9 -8 The Pythagorean Theorem Additional Example 2 Continued 4 Look Back The hypotenuse is the longest side of a right triangle. Since the distance from one corner of the field to the opposite corner is greater than the length of a side of the field, the answer is reasonable. Course 2

9 -8 The Insert Lesson Title Here Pythagorean Theorem Check It Out: Example 2 A rectangular field has a length of 100 yards and a width of 33 yards. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth. 1 Understand the Problem Rewrite the question as a statement. • Find the distance from one corner of the field to the opposite corner of the field. Course 2

9 -8 The Pythagorean Theorem Check It Out: Example 2 Continued List the important information: • Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles. • The segment between the two corners is the hypotenuse. • The sides of the fields are legs, and they are 33 yards long and 100 yards long. 2 Course 2 Make a Plan You can use the Pythagorean Theorem to write an equation.

9 -8 The Insert Lesson Title Here Pythagorean Theorem Check It Out: Example 2 Continued 3 Solve a 2 + b 2 = c 2 Use the Pythagorean Theorem. 332 + 1002 = c 2 Substitute for the known variables. 1089 + 10, 000 = c 2 Evaluate the powers. 11, 089 = c 2 Add. 105. 3043208 c 105. 3 c Take the square roots of both sides. Round. The distance from one corner of the field to the opposite corner is about 105. 3 yards. Course 2

9 -8 The Insert Lesson Title Here Pythagorean Theorem Check It Out: Example 2 Continued 4 Look Back The hypotenuse is the longest side of a right triangle. Since the distance from one corner of the field to the opposite corner is greater than the length of either side of the field, the answer is reasonable. Course 2

9 -8 The Insert Pythagorean Lesson Title Theorem Here Lesson Quiz: Part I Use the Pythagorean Theorem to find each missing measure. 1. 3. a = 40 m 21 in. , b = 30, c = 34 16 4. a = 20, b = 21, c = Course 2 2. 29

9 -8 The Insert Pythagorean Lesson Title Theorem Here Lesson Quiz: Part II 5. Each rectangular section of a fence is braced by a board nailed on the diagonal of the section. The fence is 6 ft tall and the brace is 10 ft long. What is the length of the section? 8 ft Course 2