Preview Warm Up California Standards Lesson Presentation

Warm Up Estimate each square root to nearest hundredth. 1. √30 2. √14 3. √55 4. √48 5.48 3.74 7.42 6.93

MG3.3 Know and understand the Pythagorean theorem and its converse and use it find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement. California Standards

Vocabulary Pythagorean Theorem leg hypotenuse

The Pythagorean Theorem shows that a special relationship exists between the sides of a right triangle. You can use the theorem to find the length of any side of a right triangle.

Use the Pythagorean Theorem to find the missing measure. Additional Example 1A: Calculating the Length of a Side of a Right Triangle Use the Pythagorean Theorem to find the missing measure. c 12 cm 16 cm a2 + b2 = c2 Use the Pythagorean Theorem. 122 + 162 = c2 Substitute for a and b. 144 + 256 = c2 Evaluate the powers. 400 = c2 Add. √400 = √c2 Take the square root of both sides. 20 = c The length of the hypotenuse is 20 cm.

Use the Pythagorean Theorem to find the missing measure. Additional Example 1B: Calculating the Length of a Missing Side of a Right Triangle Use the Pythagorean Theorem to find the missing measure. b 5 cm 13 cm a2 + b2 = c2 Use the Pythagorean Theorem. 52 + b2 = 132 Substitute for a and c. 25 + b2 = 169 Evaluate the powers. –25 –25 Subtract 25 from each side. b2 = 144 √b2 = √144 Take the square root of both sides. b = 12 The length of the missing leg is 12 cm.

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. 75ft 75ft 75ft 75ft

Understand the Problem 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.

Additional Example 2 Continued Make a Plan You can use the Pythagorean Theorem to write an equation.

Additional Example 2 Continued Solve 3 a2 + b2 = c2 Use the Pythagorean Theorem. 752 + 752 = c2 Substitute for the known variables. 5,625 + 5,625 = c2 Evaluate the powers. 11,250 = c2 Add. 106.066  c Take the square roots of both sides. 106.1  c Round. The distance from one corner of the field to the opposite corner is about 106.1 feet.

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.

Additional Example 3A: Identifying a Right Triangle Tell whether the given side lengths form a right triangle. A. 12, 35, 37 ? a2 + b2 = c2 Compare a2 to b2 to c2. ? 122 + 352 = 372 Substitute the longest side length for c. ? 144 + 1225 = 1369 Simplify the powers. 1369 = 1369  Add. The side lengths form a right triangle.

Additional Example 3B: Identifying a Right Triangle Tell whether the given side lengths form a right triangle. B. 8, 12, 16 ? a2 + b2 = c2 Compare a2 to b2 to c2. ? 82 + 122 = 162 Substitute the longest side length for c. ? 64 + 144 = 256 Simplify the powers. 208 ≠ 256  Add. The side lengths do not form a right triangle.

Lesson Quiz Use the Pythagorean Theorem to find each missing measure. 1. 2. 21 in. 40 m 3. 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 Tell whether the given side lengths form a right triangle. 4. 2.5, 3, 4.5 no 5. 33, 56, 65 yes