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2/17/2016 Five-Minute Check (over Lesson 8–1) CCSS Then/Now New Vocabulary Theorem 8.4: Pythagorean Theorem Proof: Pythagorean Theorem Example 1: Find Missing Measures Using the Pythagorean Theorem Key Concept: Common Pythagorean Triples Example 2: Use a Pythagorean Triple Example 3: Standardized Test Example: Use the Pythagorean Theorem Theorem 8.5: Converse of the Pythagorean Theorem Theorems: Pythagorean Inequality Theorems Example 4: Classify Triangles 1 2/17/2016 Over Lesson 8–1 Find the geometric mean between 9 and 13. Find the geometric mean between Find the altitude a. Find x, y, and z. Over Lesson 8–1 Find the geometric mean between 9 and 13. A. 2 B. 4 C. D. 2 2/17/2016 Over Lesson 8–1 Find the geometric mean between A. B. C. D. Over Lesson 8–1 Find the altitude a. A. 4 B. C. 6 D. 3 2/17/2016 Over Lesson 8–1 Find x, y, and z to the nearest tenth. A. x = 6, y = 8, z = 12 B. x = 7, y = 8.5, z = 15 C. x = 8, y ≈ 8.9, z ≈ 17.9 D. x = 9, y ≈ 10.1, z = 23 Over Lesson 8–1 Which is the best estimate for m? A. 9 B. 10.8 C. 12.3 D. 13 4 2/17/2016 Content Standards G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.MG.3 Apply geometric methods to solve problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). Mathematical Practices 1 Make sense of problems and persevere in solving them. 4 Model with mathematics. You used the Pythagorean Theorem to develop the Distance Formula. • Use the Pythagorean Theorem. • Use the Converse of the Pythagorean Theorem. 5 2/17/2016 • Pythagorean triple 6 2/17/2016 Find Missing Measures Using the Pythagorean Theorem A. Find x. Eureka! The side opposite the right angle is the hypotenuse. (leg)2 + (leg)2 = (hypotenuse)2 42 + 72 = x2 65 = x2 = 65 Find Missing Measures Using the Pythagorean Theorem B. Find x. The hypotenuse is 12 (leg)2 + (leg)2= (hypotenuse)2 x2 + 82 = 122 x2 + 64 = 144 x2 = 80 7 2/17/2016 A. Find x. A. B. C. D. B. Find x. A. B. C. D. 8 2/17/2016 To build your own Pythagorean triplet: If m > n and both m & n are natural numbers then leg = m2 – n2 , leg = 2mn and hypotenuse = m2 + n2 Use a Pythagorean Triple Use a Pythagorean triple to find x. Explain your reasoning. Notice: 2 x 13 = So: = 2 x 5 = 10 2 x 12 = Recall: 5,12,13 is a Pythagorean triplet Answer: x = 10 Check: 242 + 102 = 262 ? 676 = 676 9 2/17/2016 Use a Pythagorean triple to find x. A. 10 B. 15 C. 18 D. 24 Use the Pythagorean Theorem A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder? A3 B4 Method 1 C 12 D 15 Use a Pythagorean triple. Notice: So x = 3 x 4 = 12 5x4= =4x4 This is a 3,4,5 ∆ with a scalar of 4 = 3 x 4 = 12 10 2/17/2016 Use the Pythagorean Theorem A 20-foot ladder is placed against a building to reach a window that is 16 feet above the ground. How many feet away from the building is the bottom of the ladder? A3 B4 Method 2 C 12 D 15 Use Pythagorean theorem + + 16 = = 20 = 20 − 16 = 400 − 256 = 144 = ±12 = 12 A 10-foot ladder is placed against a building. The base of the ladder is 6 feet from the building. How high does the ladder reach on the building? A. 6 ft B. 8 ft C. 9 ft D. 10 ft 11 2/17/2016 12 2/17/2016 Classify Triangles A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem. 9 + 12 > 15 9 + 15 > 12 12 + 15 > 9 The side lengths 9, 12, and 15 can form a triangle. Classify Triangles A. Determine whether 9, 12, and 15 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 2 9 Classify the triangle + ? + 12 ? 15 81 + 144 ? 225 225 = 225 Answer: Since + = , the triangle is a right triangle. 13 2/17/2016 Classify Triangles B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 1 Determine whether the measures can form a triangle using the Triangle Inequality Theorem. 10 + 11 > 13 10 + 13 > 11 11 + 13 > 10 The side lengths 10, 11, and 13 can form a triangle. Classify Triangles B. Determine whether 10, 11, and 13 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. Step 2 10 Classify the triangle. + ? + 11 ? 13 100 + 121 ? 169 221 > 169 Answer: Since + > , the triangle is an acute triangle. 14 2/17/2016 A. Determine whether the set of numbers 7, 8, and 14 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. A. yes, acute B. yes, obtuse C. yes, right D. not a triangle B. Determine whether the set of numbers 14, 18, and 33 can be the measures of the sides of a triangle. If so, classify the triangle as acute, right, or obtuse. Justify your answer. A. yes, acute B. yes, obtuse C. yes, right D. not a triangle 15 2/17/2016 16