The Pythagorean Theorem Objectives Use the Pythagorean Theorem

The Pythagorean Theorem Objectives Use the Pythagorean Theorem and its converse to solve problems. Use Pythagorean inequalities to classify triangles. Holt Mc. Dougal Geometry

The Pythagorean Theorem Vocabulary Pythagorean triple Holt Mc. Dougal Geometry

The Pythagorean Theorem is probably the most famous mathematical relationship. As you learned in Lesson 1 -6, it states that in a right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. a 2 + b 2 = c 2 Holt Mc. Dougal Geometry

The Pythagorean Theorem Example 1 A: Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 Pythagorean Theorem 22 + 62 = x 2 Substitute 2 for a, 6 for b, and x for c. 40 = x 2 Simplify. Find the positive square root. Simplify the radical. Holt Mc. Dougal Geometry

The Pythagorean Theorem Example 1 B: Using the Pythagorean Theorem Find the value of x. Give your answer in simplest radical form. a 2 + b 2 = c 2 Pythagorean Theorem (x – 2)2 + 42 = x 2 Substitute x – 2 for a, 4 for b, and x for c. x 2 – 4 x + 4 + 16 = x 2 Multiply. – 4 x + 20 = 0 Combine like terms. 20 = 4 x Add 4 x to both sides. 5=x Holt Mc. Dougal Geometry Divide both sides by 4.

The Pythagorean Theorem Check It Out! Example 2 What if. . . ? According to the recommended safety ratio of 4: 1, how high will a 30 -foot ladder reach when placed against a wall? Round to the nearest inch. Let x be the distance in feet from the foot of the ladder to the base of the wall. Then 4 x is the distance in feet from the top of the ladder to the base of the wall. Holt Mc. Dougal Geometry

The Pythagorean Theorem Check It Out! Example 2 Continued a 2 + b 2 = c 2 (4 x)2 + x 2 = 302 17 x 2 = 900 Pythagorean Theorem Substitute 4 x for a, x for b, and 30 for c. Multiply and combine like terms. Since 4 x is the distance in feet from the top of the ladder to the base of the wall, 4(7. 28) 29 ft 1 in. Holt Mc. Dougal Geometry

The Pythagorean Theorem A set of three nonzero whole numbers a, b, and c such that a 2 + b 2 = c 2 is called a Pythagorean triple. Holt Mc. Dougal Geometry

The Pythagorean Theorem The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths. Holt Mc. Dougal Geometry

The Pythagorean Theorem You can also use side lengths to classify a triangle as acute or obtuse. B c A Holt Mc. Dougal Geometry a b C

The Pythagorean Theorem To understand why the Pythagorean inequalities are true, consider ∆ABC. Holt Mc. Dougal Geometry

The Pythagorean Theorem Remember! By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater than the third side length. Holt Mc. Dougal Geometry

The Pythagorean Theorem Example 4 A: Classifying Triangles Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 7, 10 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 5, 7, and 10 can be the side lengths of a triangle. Holt Mc. Dougal Geometry

The Pythagorean Theorem Example 4 A Continued Step 2 Classify the triangle. c 2 102 ? = a 2 + b 2 ? = 52 + 72 ? Compare c 2 to a 2 + b 2. Substitute the longest side for c. 100 = 25 + 49 Multiply. 100 > 74 Add and compare. Since c 2 > a 2 + b 2, the triangle is obtuse. Holt Mc. Dougal Geometry

The Pythagorean Theorem Example 4 B: Classifying Triangles Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 5, 8, 17 Step 1 Determine if the measures form a triangle. Since 5 + 8 = 13 and 13 > 17, these cannot be the side lengths of a triangle. Holt Mc. Dougal Geometry

The Pythagorean Theorem Check It Out! Example 4 a Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 7, 12, 16 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 7, 12, and 16 can be the side lengths of a triangle. Holt Mc. Dougal Geometry

The Pythagorean Theorem Check It Out! Example 4 a Continued Step 2 Classify the triangle. c 2 162 ? = a 2 + b 2 ? = 122 + 72 ? Compare c 2 to a 2 + b 2. Substitute the longest side for c. 256 = 144 + 49 Multiply. 256 > 193 Add and compare. Since c 2 > a 2 + b 2, the triangle is obtuse. Holt Mc. Dougal Geometry

The Pythagorean Theorem Check It Out! Example 4 b Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 11, 18, 34 Step 1 Determine if the measures form a triangle. Since 11 + 18 = 29 and 29 > 34, these cannot be the sides of a triangle. Holt Mc. Dougal Geometry

The Pythagorean Theorem Check It Out! Example 4 c Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right. 3. 8, 4. 1, 5. 2 Step 1 Determine if the measures form a triangle. By the Triangle Inequality Theorem, 3. 8, 4. 1, and 5. 2 can be the side lengths of a triangle. Holt Mc. Dougal Geometry

The Pythagorean Theorem Check It Out! Example 4 c Continued Step 2 Classify the triangle. c 2 5. 22 ? = a 2 + b 2 ? = 3. 82 + 4. 12 Compare c 2 to a 2 + b 2. Substitute the longest side for c. ? 27. 04 = 14. 44 + 16. 81 Multiply. 27. 04 < 31. 25 Add and compare. Since c 2 < a 2 + b 2, the triangle is acute. Holt Mc. Dougal Geometry