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Unit 6 – Right Triangles and Trigonometry INVESTIGATION 1 PRACTICE PROBLEMS
The Pythagorean Theorem and Right Triangles
I can use the Pythagorean Theorem, the Converse of the Pythagorean
Theorem, classify and solve problems involving triangles.
Lesson Practice Problem Options
Max Possible Points
Total Points Earned
1: The Pythagorean Theorem
#1, 2, 3, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 36, 37,
38, 39, 41
18 points
2: Special Right Triangles #1, 2, 6, 8, 10, 12, 14, 16, 18, 20, 21,
24, 26, 27 11 points
3: Similar Right Triangles
#1, 2, 6, 8, 10, 12, 14, 16, 18, 20,
22, 24, 26, 28, 30, 32, 38, 39, 40, 41, 46, 47,
48, 49
19 points
________/48 points
** In order to earn credit for practice problems, ALL WORK must be shown.**
Lesson 1: The Pythagorean Theorem
1. What is a Pythagorean triple?
2. Which is different? Find “both answers”
In Exercises 3, 4, 6, and 10 find the value of x. Then tell whether the side lengths form a Pythagorean triple. 3. 4.
.
6. .
10. .
ERROR ANALYSIS. In Exercise 12, describe and correct the error in using the Pythagorean Theorem.
12. .
14. MODELING WITH MATHEMATICS. The backboard of the basketball hoop forms a right triangle with the supporting rods, as shown. Use the Pythagorean Theorem to appropriate the distance between the rods where they meet the backboard.
In Exercises 16, 18 ad 20, tell whether the triangle is a right triangle.
16. .
18. .
20. In Exercises 22, 24, 26 and 28, verify that the segment lengths form a triangle. Is the triangle acute, right, or obtuse?
22. 6, 8 and 10.
24. 15, 20, and 36
26. 4.1, 8.2, and 12.2
28. 10, 15, and 5√13
30. REASONING. You ar making a canvas frame for a painting using stretcher bars. The rectangular painting will be 10 inches long and 8 inches wide. Using a ruler, how can you be certain that the corners of the frame are 90℉?
32. Find the area of the isosceles triangle.
36. HOW DO YOU SEE IT? How do you know angle C is a right angle without using the Pythagorean Theorem?
37. PROBLEM SOLVING. You are making a kite and need to figure out how much binding to buy. You need the
binding for the perimeter of the kite. The binding comes in packages of two yards. How many packages should
you buy?
38. PROVING A THEOREM. Use the Pythagorean Theorem to prove the Hypotenuse-Leg (HL) Congruence Theorem.
39. PROVING A THEOREM. Prove the Converse of the Pythagorean Theorem. (Hint: Draw ∆𝐴𝐵𝐶 with side lengths a,
b, and c, where c is the length of the longest side. Then draw a right triangle with side lengths a, b, and x, where
x is the length of the hypotenuse. Compare lengths c and x.)
41. MAKING AN ARGUMENT. Your friend claims 72 and 75 cannot be part of a Pythagorean triple because 722 + 752 does not equal a positive integer squared. Is your friend correct? Explain your reasoning.
2: Special Right Triangles
1. VOCABULARY. Name two special right triangles by their angle measures.
2. WRITING. Explain why the acute angles in an isosceles right triangle always measure 45°.
6. Find the value of x. Write your answer in simplest form.
For questions 8 and 10, find the values of x and y. Write your answers in simplest form.
8.
10.
12. Describe and correct the error in finding the length of the hypotenuse.
14. Sketch a square that has a perimeter of 36 inches. Find the length of a diagonal (Round your answer to the nearest tenth).
16. Find the area of the figure (round your answer to the nearest tenth).
18. MODELING WITH MATHEMATICS. A nut is shaped like a regular hexagon with side lengths of 1 centimeter. Find the value of x. (Hint: A regular hexagon can be divided into six congruent triangles.)
20. HOW DO YOU SEE IT? The diagram shows part of the Wheel of Theodorus.
a. Which triangles, if any, are 45° − 45° − 90° triangles?
b. Which triangles, if any, are 30° − 60° − 90° triangles?
21. PROVING A THEOREM. Write a paragraph proof of the 30° − 60° − 90° Triangle Theorem. (Hint: Construct ∆𝐽𝑀𝐿 congruent to ∆𝐽𝐾𝐿.) Given ∆𝐽𝑀𝐿 is a 30° − 60° − 90° triangle. Prove The hypotenuse is twice as long as the shorter leg, and the longer leg is
√3 times as long as the shorter leg.
24. MAKING AN ARGUMENT. Each triangle in the diagram is a 45° − 45° − 90° triangle. At Stage 0, the legs of the triangle are each 1 unit long. Your brother claims the lengths of the legs of the triangles added are halved at each stage. So, the length of a leg of a triangle added in
Stage 8 will be 1
256 unit. Is your brother correct? Explain
your reasoning. For questions 26 and 27, find the value of x.
26. ∆𝐷𝐸𝐹~∆𝐿𝑀𝑁
27. ∆𝐴𝐵𝐶~∆𝑄𝑅𝑆
3: Similar Right Triangles
1. COMPLETE THE SENTENCE. If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles
formed are similar to the origianl triangle and ______________________.
2. WRITING. In your own words, explain geometric mean.
For questions 6, 8 and 10, find the value of x.
6. .
8. .
10. .
For problems 12, 14, 16 and 18, find the geometric mean of the two numbers.
12. 9 and 16
14. 25 and 35
16. 8 and 28
18. 24 and 45
For problems 20, 22, 24 and 26, find the variable.
20. .
22. .
24. .
26. .
28. Describe and correct the error in writing an equation for the given diagram.
30. Your classmate is standing on the other side of the monument. She has a piece of rope staked at the base of the monument. She extends the rope to the cardboard square she is holding lined up to the top and bottom of the monument. Use the information in the diagram above to approximate the height of the monument.
32. Find the value of the variable.
38. HOW DO YOU SEE IT? In which of the following trianlges does the Geometric Mean (Altitude) Theorem apply?
39. PROVING A THEOREM. Use the diagram of ∆𝐴𝐵𝐶. Complete the proof of the Pythagorean Theorem.
40. MAKING AN ARGUMENT. Your friend claims the geometric mean of 4 and 9 is 6, and then labels the triangle, as
shown. Is your friend correct? Explain your reasoning.
41. Use the given statements to prove the theorem.
Prove the Geometric Mean (Altitude Theorem) by showing that 𝐶𝐷2 = 𝐴𝐷 ∙ 𝐵𝐷.
Solve the equation for x.
