Unit 2. Looking for Pythagoras. Investigation 5: Using the Pythagorean Theorem: Analyzing Triangles and Circles
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1 I can understand and apply the Pythagorean Theorem. Investigation 5 Unit 2 Looking for Pythagoras Investigation 5: Using the Pythagorean Theorem: Analyzing Triangles and Circles Lesson 1: Stopping Sneaky Sally (Finding Unknown Side Practice Problems Lengths) #1-4 Lesson 2: Analyzing Triangles In this Investigation, you will apply the Pythagorean Theorem in some very different situations. Whenever there is a right triangle in a figure, you can use the Pythagorean Theorem to deduce the side lengths of the triangle. Sometimes the triangle is not obvious. Lesson 1: Stopping Sneaky Sally (Finding Unknown Side Lengths) I can Understand and apply the Pythagorean Theorem. A baseball diamond is actually a square. If you can find right trianlges in this shape, you can use the Pythagorean Theorem to solve problems about distances. Problem 5.1 Horace Hanson is the catcher for the Humboldt Bees baseball team. Sneaky Sally Smith, the star of the Canfield Cats, is on first base. Sally is known for stealing bases, so Horace is keeping an eye on her. The pitcher throws a fastball, and the batter swings and misses. Horace catches the pitch and, out of the corner of his eye, he sees Sally take off for second base.
2 Use the diagram to answer Questions A C. A. How far must Horace throw the baseball to get Sally out at second base? Explain. 1. Jen says the distance that Horace throws the baseball is a rational number. Florence says that it is an irrational number. Explain each student s reasoning. B. The shortstop is standing on the baseline, halfway between second base and third base. How far is the shortstop from Horace?
3 C. The pitcher s mound is 60 feet 6 inches from home plate. Use this information and your answer to Question A to find the distance from the pitcher s mound to each base. Lesson 2: Analyzing Triangles I can understand and apply the Pythagorean Theorem. You can use the Pythagorean Theorem to investigate some interesting properties of an equilateral triangle. One property is that all equilateral triangles have reflection symmetries. Triangle ABC is an equilateral triangle. Line AP is a reflection line for triangle ABC. If you fold an equilateral triangle along the line of reflection, you will find some properties of any equilateral triangle. What is true about the angle measures of an equilateral triangle? What is true about the side lengths of an equilateral triangle? What can you say about the measures of angle CAP, angle BAP, angle CPA, and angle BPA? What can you say about line segments CP and BP? What can you say about triangles ACP and ABP?
4 Is there a relationship among the lengths of line segments CP, AP, and AC? Problem 5.2 A. Suppose the lengths of the sides of equilateral triangle ABC are 2 units. Identify the following measures: 1. angle CAP 5. length of CP 2. angle BAP 6. length of BP 3. angle CPA 7. length of AP 4. angle BPA B. Suppose the lengths of the sides of equilateral triangle ABC are 4 units. Identify the following measures: 1. angle CAP 5. length of CP 2. angle BAP 6. length of BP 3. angle CPA 7. length of AP 4. angle BPA
5 C. Thomas thinks he has a way of predicting the length of the height AP for any equilateral triangle. He has drawn the results of Questions A and B in the diagram at the right. 1. The triangles look similar. Are they? Explain. 2. What is the length of A 2P? What is the length of C 2P? 3. Is the length of A 2P the same as the length of AP you found in Question B? Explain. D. A right triangle with a 60 angle is called a triangle. The triangle at the right has a hypotenuse of length 10 units. 1. What are the lengths of the other two sides? Explain how you found your answers. 2. What relationship among the side lengths do you observe for this triangle? Is this relationship true for all triangles? Explain.
6 3. If the hypotenuse of a triangle is s units long, what are the lengths of the other two sides? (This is your formula for calculating side lengths in a triangle.) E. Use the figure below. 1. How many right triangles do you see in the figure? 2. Find the perimeter of triangle ABC. Explain your strategy. 3. Find the area of triangle ABC. Explain your strategy.
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