Five-Minute Check (over Lesson 3-4) Main Idea and Vocabulary Targeted TEKS Key Concept: Pythagorean Theorem Example 1: Find the Length of a Side Example 2: Find the Length of a Side Key Concept: Converse of Pythagorean Theorem Example 3: Identify a Right Triangle Lesson 3-5 Menu

Lesson 3-5 Ideas/Vocabulary Use the Pythagorean Theorem. legs hypotenuse Pythagorean Theorem converse Lesson 3-5 Ideas/Vocabulary

8.7 The student uses geometry to model and describe the physical world. (C) Use pictures or models to demonstrate the Pythagorean Theorem. 8.9 The student uses indirect measurement to solve problems. (A) Use the Pythagorean Theorem to solve real-life problems. Lesson 3-5 TEKS

Lesson 3-5 Key Concept 1

Find the Length of a Side Write an equation to find the length of the missing side of the right triangle. Then find the missing length. Round to the nearest tenth, if necessary. Lesson 3-5 Example 1

Find the Length of a Side c2 = a2 + b2 Pythagorean Theorem c2 = 122 + 162 Replace a with 12 and b with 16. c2 = 144 + 256 Evaluate 122 and 162. c2 = 400 Add 144 and 256. c = Definition of square root c = 20 or –20 Simplify. Answer: The equation has two solutions, 20 and –20. However, the length of a side must be positive. So, the hypotenuse is 20 inches long. Lesson 3-5 Example 1

Write an equation to find the length of the missing side of the right triangle. Then find the missing length. Round to the nearest tenth, if necessary. A B C D A. 17 in. B. 19 in. C. 20 in. D. 21 in. Lesson 3-5 Example 1 CYP

Find the Length of a Side The hypotenuse of a right triangle is 33 centimeters long and one of its legs is 28 centimeters. What is a, the length of the other leg? c2 = a2 + b2 Pythagorean Theorem 332 = a2 + 282 Replace c with 33 and b with 28. 1,089 = a2 + 784 Evaluate 332 and 282. 1,089 – 784 = a2 + 784 – 784 Subtract 784 from each side. 305 = a2 Simplify. = a Definition of square root 17.5 ≈ a Use a calculator. Lesson 3-5 Example 2

Find the Length of a Side Answer: The length of the other leg is about 17.5 centimeters. Lesson 3-5 Example 2

The hypotenuse of a right triangle is 26 centimeters long and one of its legs is 17 centimeters. What is a, the length of the other leg? A B C D A. about 16.2 cm B. about 18.5 cm C. about 19.7 cm D. about 21.4 cm Lesson 3-5 Example 2 CYP

Lesson 3-5 Key Concept 2

Identify a Right Triangle The measures of three sides of a triangle are 24 inches, 7 inches, and 25 inches. Determine whether the triangle is a right triangle. c2 = a2 + b2 Pythagorean Theorem 252 = 72 + 242 Replace a with 7, b with 24, and c with 25. ? ? 625 = 49 + 576 Evaluate 252, 72, and 242. 625 = 625 Simplify. Answer: The triangle is a right triangle. Lesson 3-5 Example 3

The measures of three sides of a triangle are 13 inches, 5 inches, and 12 inches. Determine whether the triangle is a right triangle. A. It is a right triangle. B. It is not a right triangle. C. Not enough information to determine. A B C Lesson 3-5 CYP 3

End of Lesson 3-5

Five-Minute Check (over Lesson 3-5) Main Idea Targeted TEKS Example 1: Use the Pythagorean Theorem to Solve a Problem Example 2: Test Example Lesson 3-6 Menu

Lesson 3-6 Ideas/Vocabulary Solve problems using the Pythagorean Theorem. Lesson 3-6 Ideas/Vocabulary

8.7 The student uses geometry to model and describe the physical world. (C) Use pictures or models to demonstrate the Pythagorean Theorem. 8.9 The student uses indirect measurement to solve problems. (A) Use the Pythagorean Theorem to solve real-life problems. Lesson 3-6 TEKS

Use the Pythagorean Theorem to Solve a Problem RAMPS A ramp to a newly constructed building must be built according to the guidelines stated in the Americans with Disabilities Act. If the ramp is 24.1 feet long and the top of the ramp is 2 feet off the ground, how far is the bottom of the ramp from the base of the building? Notice the problem involves a right triangle. Use the Pythagorean Theorem. Lesson 3-6 Example 1

Use the Pythagorean Theorem to Solve a Problem 24.12 = a2 + 22 Replace c with 24.1 and b with 2. 580.81 = a2 + 4 Evaluate 24.12 and 22. 580.81 – 4 = a2 + 4 – 4 Subtract 4 from each side. 576.81 = a2 Simplify. = a Definition of square root 24.0 ≈ a Simplify. Answer: The end of the ramp is about 24 feet from the base of the building. Lesson 3-6 Example 1

RAMPS If a truck ramp is 32 feet long and the top of the ramp is 10 feet off the ground, how far is the end of the ramp from the truck? A B C D A. about 30.4 feet B. about 31.5 feet C. about 33.8 feet D. about 35.1 feet Lesson 3-6 Example 1 CYP

Use the Pythagorean Theorem The cross-section of a camping tent is shown below. Find the width of the base of the tent. A. 6 ft B. 8 ft C. 10 ft D. 12 ft Lesson 3-6 Example 2

Use the Pythagorean Theorem Read the Test Item From the diagram, you know that the tent forms two congruent right triangles. Let a represent half the base of the tent. Then w = 2a. Lesson 3-6 Example 2

Use the Pythagorean Theorem Solve the Test Item Use the Pythagorean Theorem. c2 = a2 + b2 Write the relationship. 102 = a2 + 82 c = 10 and b = 8 100 = a2 + 64 Evaluate 102 and 82. 100 – 64 = a2 + 64 – 64 Subtract 64 from each side. 36 = a2 Simplify. = a Definition of square root 6 = a Simplify. Lesson 3-6 Example 2

Use the Pythagorean Theorem The cross-section of a camping tent is shown below. Find the width of the base of the tent. A. 6 ft B. 8 ft C. 10 ft D. 12 ft Answer: The width of the base of the tent is 2a or (2)6 = 12 feet. Therefore, choice D is correct. Lesson 3-6 Example 2

This picture shows the cross-section of a roof This picture shows the cross-section of a roof. How long is each rafter, r? A B C D A. 15 ft B. 18 ft C. 20 ft D. 22 ft Lesson 3-6 Example 2 CYP

End of Lesson 3-6

Five-Minute Check (over Lesson 3-6) Main Ideas and Vocabulary Targeted TEKS Example 1: Name an Ordered Pair Example 2: Name an Ordered Pair Example 3: Graphing Ordered Pairs Example 4: Graphing Ordered Pairs Example 5: Find Distance on the Coordinate Plane Example 6: Use a Coordinate Plane to Solve a Problem Lesson 3-7 Menu

Lesson 3-7 Ideas/Vocabulary Graph rational numbers on the coordinate plane. Find the distance between two points on the coordinate plane. coordinate plane ordered pair x-coordinate abscissa y-coordinate ordinate origin y-axis x-axis quadrants Lesson 3-7 Ideas/Vocabulary

8.7 The student uses geometry to model and describe the physical world. (D) Locate and name points on the coordinate plane using ordered pairs of rational numbers. 8.9 The student uses indirect measurement to solve problems. (A) Use the Pythagorean Theorem to solve real-life problems. Also addresses TEKS 8.1(C). Lesson 3-7 TEKS

Name the ordered pair for point A. Name an Ordered Pair Name the ordered pair for point A. Start at the origin. Move right to find the x-coordinate of point A, which is 2. Move up to find the y-coordinate, which is Answer: So, the ordered pair for point A is Lesson 3-7 Example 1

Name the ordered pair for point A. B C D A. B. C. D. Lesson 3-7 Example 1 CYP

Name the ordered pair for point B. Name an Ordered Pair Name the ordered pair for point B. Start at the origin. Move left to find the x-coordinate of point B, which is Move down to find the y-coordinate, which is –2. Answer: So, the ordered pair for point B is Lesson 3-7 Example 2

Name the ordered pair for point B. C D A. B. C. D. Lesson 3-7 Example 2 CYP

Graphing Ordered Pairs Graph and label point J(–3, 2.75). Start at the origin and move 3 units to the left. Then move up 2.75 units. Draw a dot and label it J(–3, 2.75). Answer: Lesson 3-7 Example 3

Graph and label point J(–2.5, 3.5). C. D. A B C D Lesson 3-7 Example 3 CYP

Graphing Ordered Pairs Graph and label point K Start at the origin and move 4 units to the right. Then move down units. Answer: Draw a dot and label it K Lesson 3-7 Example 4

Graph and label point K A. B. C. D. A B C D Lesson 3-7 Example 4 CYP

Find Distance in the Coordinate Plane Graph the ordered pairs (0, –6) and (5, –1). Then find the distance between the points. Let c = the distance between the two points, a = 5, and b = 5. Lesson 3-7 Example 5

Find Distance in the Coordinate Plane c2 = a2 + b2 Pythagorean Theorem c2 = 52 + 52 Replace a with 5 and b with 5. c2 = 50 52 + 52 = 50 = Definition of square root c ≈ 7.1 Simplify. Answer: The points are about 7.1 units apart. Lesson 3-7 Example 5

Graph the ordered pairs (0, –3) and (2, –6) Graph the ordered pairs (0, –3) and (2, –6). Then find the distance between the points. A B C D A. about 3.1 units B. about 3.6 units C. about 3.9 units D. about 4.2 units Lesson 3-7 Example 5 CYP

Use a Coordinate Plane to Solve a Problem TRAVEL Melissa lives in Chicago, Illinois. A unit on the grid of her map shown below is 0.08 mile. Find the distance between McCormickville at (–2, –1) and Lake Shore Park at (2, 2). Let c = the distance between McCormickville and Lake Shore Park. Then a = 3 and b = 4. Lesson 3-7 Example 6

Use a Coordinate Plane to Solve a Problem c2 = a2 + b2 Pythagorean Theorem c2 = 32 + 42 Replace a with 3 and b with 4. c2 = 25 32 + 42 = 25 = Definition of square root c = 5 Simplify. The distance between McCormickville and Lake Shore Park is 5 units on the map. Answer: Since each unit equals 0.08 mile, the distance is 0.08  5 or 0.4 mile. Lesson 3-7 Example 6

TRAVEL Sato lives in Chicago TRAVEL Sato lives in Chicago. A unit on the grid of his map shown below is 0.08 mile. Find the distance between Shantytown at (2, –1) and the intersection of N. Wabash Ave. and E. Superior St. at (–3, 1). A B C D A. about 0.1 mile B. about 0.2 mile C. about 0.3 mile D. about 0.4 mile Lesson 3-7 Example 6 CYP

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