H Geo Unit 7 Student Packet - First Half PDF

# CRM Unit 7: Right Triangle Trigonometry ## Mrs. Gunby ### Name: ### Important Dates: ### Period: ## Topic: ### Main Ideas/Questions ### Notes/Examples #### PYTHAGOREAN THEOREM - Used to find the missing *hypotenuse* of a *right* triangle. - Sides *a* and *b* are called *legs*. - Side *c* is cal...

# CRM Unit 7: Right Triangle Trigonometry ## Mrs. Gunby ### Name: ### Important Dates: ### Period: ## Topic: ### Main Ideas/Questions ### Notes/Examples #### PYTHAGOREAN THEOREM - Used to find the missing *hypotenuse* of a *right* triangle. - Sides *a* and *b* are called *legs*. - Side *c* is called the *hypotenuse*. - For any right triangle: $a^2 + b^2 = c^2$ #### Directions: Find the value of *x*. Round each answer to the nearest tenth. | Example | Diagram | *x* | |---|---|---| | 1. | A right triangle with legs of length 8 and 13, hypotenuse of length *x*. | 15 | | 2. | A right triangle with legs of length 22 and 27, hypotenuse of length *x*. | 34.5 | | 3. | A right triangle with legs of length 7 and 9, hypotenuse of length *x*. | 11.4 | | 4. | A right triangle with legs of length *x* and 30.5, hypotenuse of length 19.1 | 11.8 | | 5. | A right triangle with legs of length 22 and 24, hypotenuse of length *x*. | 32.2 | | 6. | A right triangle with legs of length 5 and 16, hypotenuse of length *x*. | 16.8 | | 7. | A right triangle with legs of length 12 and *x*, hypotenuse of length 29. | 24.2 | | 8. | A right triangle with legs of length 10 and 25, hypotenuse of length 28. | 17.9 | | 9. | A right triangle with legs of length 20 and *x*, hypotenuse of length 26. | 15.6 | ## Applications #### Directions: Draw a picture, then solve for the missing side. 10. A roofer leaned a 16-foot ladder against a house. If the base of the ladder is 5 feet from the house, how high up the house does the ladder reach? 11. Kurt is building a rectangular deck. If the dimensions of the deck are 10 feet by 23 feet, what is the length of the diagonal of the deck? 12. Ashley jogged 3.4 miles east, then 5.7 miles south. How far is Ashley from her starting point? 13. A 31-foot support wire is attached from the top of a 25 foot telephone pole to a point on the ground. How far from the base of the pole does the wire meet the ground? # **PYTHAGOREAN THEOREM CONVERSE** ## Topic ### Main Ideas/Questions ### Notes/Examples Given a triangle with sides a, b, and c: - If $a^2 + b^2 = c^2$, then the triangle is *right*. - If $a^2 + b^2 < c^2$, then the triangle is *obtuse*. - If $a^2 + b^2 > c^2$, then the triangle is *acute*. #### Directions: First, determine if the three side lengths could form a triangle. *Recall from earlier, the sum of the two smaller sides must be greater than the third side)*. If yes, classify the triangle further as acute, right, or obtuse. | Example | Side Lengths | Classification | |---|---|---| | 1. | 3, 7, 9 | Acute | | 2. | 20, 21, 29 | Right | | 3. | 4, 11, 16 | Obtuse | | 4. | 17, 17, 22 | Acute | | 5. | 18, 24, 30 | Right | | 6. | 8, 15, 23 | Obtuse | | 7. | 31, 35, 39 | Acute | | 8. | 11, 19, 28 | Obtuse | # **SPECIAL RIGHT TRIANGLES** ## Topic ### Main Idea/Questions ### Notes/Examples #### **45°-45°-90° Special Right Triangle** - Leg = *x* - Hypotenuse = *x√2* #### **Directions:** Find the value of each variable. >*Note*: The legs of a 45°-45°-90° triangle are always congruent. | Example | Diagram | x | y | |---|---|---|---| | 1. | A 45°-45°-90° triangle with a leg of length 8 and a hypotenuse of length y. | 8√2 | 8 | | 2. | A 45°-45°-90° triangle with a leg of length *x* and a hypotenuse of length 25. | 25/√2 | 25/√2 | | 3. | A 45°-45°-90° triangle with a leg of length *x* and a hypotenuse of length 19. | 19/√2 | 19/√2 | #### **30°-60°-90° Special Right Triangle** - Shorter Leg = *x* - Longer Leg = *x√3* - Hypotenuse = *2x* #### **Directions:** Find the value of each variable. >*Note*: The shorter leg is always opposite the 30° angle and the longer leg is always opposite the 60° angle. | Example | Diagram | x | y | |---|---|---|---| | 4. | A 30°-60°-90° triangle with a shorter leg of length 5 and a hypotenuse of length y. | 5 | 10 | | 5. | A 30°-60°-90° triangle with a longer leg of length 14 and a shorter leg of length *x*. | 14/√3 | 14/√3 | | 6. | A 30°-60°-90° triangle with a shorter leg of length *x*, a hypotenuse of length 32, and a longer leg of length y. | 16 | 16√3 | | 7. | A 30°-60°-90° triangle with a longer leg of length 46, a shorter leg of length *x*, and a hypotenuse of length y. | 46/√3 | 92/√3 | | 8. | A 30°-60°-90° triangle with a longer leg of length 20, a shorter leg of length *x*, and a hypotenuse of length y. | 20/√3 | 40/√3 | | 9. | A 30°-60°-90° triangle with a shorter leg of length 9√3, a longer leg of length *x*, and a a hypotenuse of length y. | 27 | 27√3 | # **MIXED PRACTICE** #### Directions: Find the value of each variable. | Example | Diagram | x | y | z | |---|---|---|---|---| | 10. | A 45°-45°-90° triangle with a hypotenuse of length 28, a leg of length *x*, and a leg of length y. | 14√2 | 14√2 | - | | 11. | A 30°-60°-90° triangle with a hypotenuse of length 16, a shorter leg of length *x*, and a longer leg of length y. | 8 | 8√3 | - | | 12. | A 45°-45°-90° triangle with a leg of length 4, a hypotenuse of length *x*, and a leg of length y. | 4√2 | 4√2 | - | | 13. | A 30°-60°-90° triangle with a hypotenuse of length 11, a shorter leg of length *x*, and a longer leg of length y. | 11/2 | 11√3 / 2 | - | | 14. | A 30°-60°-90° triangle with a shorter leg of length 15√3, a longer leg of length *x*, and a a hypotenuse of length y. | 45 | 45√3 | - | | 15. | A 45°-45°-90° triangle with a leg of length 9√2, a hypotenuse of length *x*, and a leg of length y. | 18 | 18 | - | | 16. | A 30°-60°-90° triangle with shorter leg*x*, a longer leg of length 6, and a hypotenuse of length y. | 2√3 | 4√3 | - | | 17. | A 45°-45°-90° triangle with a leg of length √10, a hypotenuse of length *x*, and a leg of length y. | 2√5 | 2√5 | - | | 18. | A 30°-60°-90° triangle with hypotenuse*x*, a longer leg of length 38, and a shorter leg of length y. | 76/√3 | 38 | - | | 19. | A 45°-45°-90° triangle with a leg of length 9, a hypotenuse of length *x*, and a leg of length y. | 9√2 | 9√2 | 30 | | 20. | A 30°-60°-90° triangle with a shorter leg of length *x*, a hypotenuse of length 32, and a longer leg of length y. | 16 | 16√3 | 16/√3 | | 21. | A 30°-60°-90° triangle with a shorter leg of length 12, a hypotenuse of length *x*, and a longer leg of length y. | 24 | 12√3 | 12/√3 | | 22. | A 30°-60°-90° triangle with a shorter leg of length 7√3, a longer leg of length *x*, and a hypotenuse of length y. | 14 | 14√3 | 7 | | 23. | A 45°-45°-90° triangle with a leg of length 45√2, a hypotenuse of length *x*, and a leg of length y. | 90 | 90 | - | | 24. | A 30°-60°-90° triangle with a shorter leg of length *x*, a longer leg of length 18, and a hypotenuse of length y. | 6√3 | 12√3 | 12 | # **SIMILARITY IN RIGHT TRIANGLES** ## Topic ### Main Ideas/Questions ### Notes/Examples #### **Right Triangle Similarity Theorem:** If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. #### **Directions:** Identify the similar triangles in the diagram, then sketch them so the corresponding sides and angles have the same orientation. 1. A right triangle with altitude *CD* drawn from the right angle *C* to the hypotenuse *AB*. Label the point where the altitude intersects the hypotenuse *D*. #### **Directions:** Find the value of *x*. | Example | Diagram | x | |---|---|---| | 2. | A right triangle with altitude drawn to the hypotenuse. The hypotenuse is of length 15. The sides that form the smaller right triangle are of lengths 9 and 12, and the sides that form the larger right triangle are of lengths *x* and 12. | 10 | | 3. | A right triangle with altitude drawn to the hypotenuse. The hypotenuse is of length 21. The sides that form the smaller right triangle are of lengths 11.2 and 23.8, and the sides that form the larger right triangle are of lengths *x* and 21. | 11.2 | | 4. | A right triangle with altitude drawn to the hypotenuse. The hypotenuse is of length 25. The sides that form the smaller right triangle are of lengths 7 and 24, and the sides that form the larger right triangle are of lengths *x* and 25. | 7 | # **GEOMETRIC MEAN** ### Notes/Examples The geometric mean of two positive numbers *a* and *b* is the positive number *x* such that: $x^2 = ab$ therefore, $ x = \sqrt{ab}$ #### **Directions:** Find the geometric mean of each pair of numbers. | Example |Numbers | Geometric Mean | |---|---|---| | 5. | 6 and 30 | √180 ≈13.4 | | 6. | 16 and 28 | √448 ≈ 21.2 | | 7. | 12 and 50 | √600 ≈ 24.5 | # **GEOMETRIC MEAN: Altitude Theorem** ### Notes/Examples **Geometry Mean (Altitude) Theorem:** The length of the altitude is the geometric mean of the lengths of the two segments. #### **Directions:** Find the value of *x*. | Example | Diagram | x | |---|---|---| | 8. | A right triangle with altitude drawn, splitting the hypotenuse into segments of lengths 8 and 12. The altitude is of length *x*. | √96 ≈ 9.8 | | 9. | A right triangle with altitude drawn, splitting the hypotenuse into segments of lengths 5 and 37. The altitude is of length *x*. | √185 ≈ 13.6 | | 10. | A right triangle with altitude drawn, splitting the hypotenuse into segments of lengths 15 and 18. The altitude is of length *x*. | √270 ≈ ‎16.4 | | 11. | A right triangle with altitude drawn, splitting the hypotenuse into segments of lengths 30 and *x*. The altitude is of length 21. | 14 | # **GEOMETRIC MEAN: Leg Theorem** ### Notes/Examples **Geometry Mean (Leg) Theorem:** The length of a leg of the triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to the leg. #### **Directions:** Find the value of *x*. | Example | Diagram | x | |---|---|---| | 12. | A right triangle with altitude drawn, splitting the hypotenuse into segments of lengths 2 and 24. One of the legs is of length *x*. | √48 ≈ 6.9 | | 13. | A right triangle with altitude drawn, splitting the hypotenuse into segments of lengths 7 and 12. One of the legs is of length *x*. | √84 ≈ 9.2 | #### **Directions:** Find the values of *x*, *y*, and *z*. | Example | Diagram | x | y | z | |---|---|---|---|---| | 14. | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 27. The two segments of the hypotenuse are of lengths *x* and *y*. The altitude is of length *z*. | 9 | 18 | 12 | | 15. | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 20. The two segments of the hypotenuse are of lengths *x* and *y*. The altitude is of length *z*. | 8 | 12 | 9.8 | # **Unit 7: Right Triangles & Trigonometry** ## Homework 1: Pythagorean Theorem and its Converse #### **Directions:** Find the value of *x*. 1. | Diagram | x | |---|---|---| | A right triangle with a leg of length 10, a leg of length 7, and a hypotenuse of length *x*.| 12.2 | 2. | Diagram | x | |---|---|---| | A right triangle with a leg of length 21, a leg of length *x*, and a hypotenuse of length 19. | 8.6 | 3. | Diagram | x | |---|---|---| | A right triangle with a leg of length 16, a leg of length *x*, and a hypotenuse of length 27. | 23.6 | 4. | Diagram | x | |---|---|---| | A right triangle with a leg of length 5.3, a leg of length *x*, and a hypotenuse of length 12.8. | 11.7 | 5. | Diagram | x | |---|---|---| | A right triangle with a leg of length *x*, a leg of length 18, and a hypotenuse of length 20. | 8.2 | 6. | Diagram | x | |---|---|---| | A right triangle with a leg of length 17, a leg of length *x*, and a hypotenuse of length 31. | 28.1 | 7. | Diagram | x | |---|---|---| | A right triangle with a leg of length 16, a leg of length 22, and a hypotenuse of length 44. | 36.9 | 8. Scott is using a 12-foot ramp to help load furniture into the back of a moving truck. If the back of the truck is 3.5 feet from the ground, what is the horizontal distance from where the ramp reaches the ground to the truck? 9. A 35-foot wire is secured from the top of a flagpole to a stake in the ground. If the stake is 14 feet from the base of the flagpole, how tall is the flagpole? 10. If the diagonal of a square is 11.3 meters, approximately what is the perimeter of the square? #### Directions: Given the side lengths, determine whether the triangle is acute, right, obtuse, or not a triangle. | Example | Side Lengths | Classification | |---|---|---| | 11. | 15, 16, 21 | Acute | | 12. | 20, 23, 41 | Not a triangle | | 13. | 10, 24, 26 | Right | | 14. | 6, 13, 20 | Obtuse | | 15. | 3, 16, 17 | Not a triangle | | 16. | 24, 29, 32 | Obtuse | # **Unit 7: Right Triangles & Trigonometry** ## Homework 2: Special Right Triangles #### **Directions:** Find the value of each variable. 1. | Diagram | x | y | |---|---|---| | A 45°-45°-90° triangle with a hypotenuse of length 13, a leg of length *x*, and a leg of length y. | 13/√2 | 13/√2 | 2. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a shorter leg of length 30, a hypotenuse of length *x*, and a longer leg of length y. | 60 | 30√3 | 3. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a shorter leg of length 3, a longer leg of length *x*, and a hypotenuse of length y. | 3√3 | 6 | 4. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a hypotenuse of length 34, a shorter leg of length *x*, and a longer leg of length y. | 17 | 17√3 | 5. | Diagram | x | y | |---|---|---| | A 45°-45°-90° triangle with a hypotenuse of length 10√2, a leg of length *x*, and a leg of length y. | 10 | 10 | 6. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a shorter leg of length *x*, a hypotenuse of length 25√3, and a longer leg of length y. | 25 | 25√3 | 7. | Diagram | x | y | |---|---|---| | A 45°-45°-90° triangle with a hypotenuse of length 2√14, a leg of length *x*, and a leg of length y. | √14 | √14 | 8. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a shorter leg of length *x*, a hypotenuse of length y, and a longer leg of length 24. | 12 | 24 | 9. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a hypotenuse of length y, a longer leg of length 22√3, and a shorter leg of length *x*. | 11 | 22 | 10. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a shorter leg of length √6, a hypotenuse of length *x*, and a longer leg of length y. | 2√6 | 2√6 *√3 | 11. | Diagram | x | y | |---|---|---| | A 45°-45°-90° triangle with a hypotenuse of length √10, a leg of length *x*, and a leg of length y. | √5 | √5 | 12. | Diagram | x | y | |---|---|---| | A 30°-60°-90° triangle with a shorter leg of length x, a hypotenuse of length 4√21, and a longer leg of length y. | 2√21 | 4√7 | 13. | Diagram | x | y | z | |---|---|---|---|---| | A 45°-45°-90° triangle with a leg of length 17, a hypotenuse of length *x*, and a leg of length y. The right angle is labeled *z*. | 17√2 | 17 | - | 14. | Diagram | x | y | z | |---|---|---|---|---| | A 30°-60°-90° triangle with a shorter leg of length *x*, a hypotenuse of length *y*. The longer leg is of length 27. The right angle is labeled *z*. | 9√3/2 | 18√3/2 | - | 15. | Diagram | x | y | z | |---|---|---|---|---| | A 45°-45°-90° triangle with a hypotenuse of length *x*, a leg of length y, and a leg of length 14√2. The right angle is labeled *z*. | 28 | 28 | - | 16. | Diagram | x | y | z | |---|---|---|---|---| | A 30°-60°-90° triangle with a shorter leg of length *x*, a hypotenuse of length *y*, and a longer leg of length 16√3. The right angle is labeled *z*. | 8√3 | 16√3 | - | 17. | Diagram | x | y | z | |---|---|---|---|---| | A 45°-45°-90° triangle with a hypotenuse of length *x*, a leg of length y, and a leg of length 39. The right angle is labeled *z*. | 39√2 | 39 | - | 18. | Diagram | x | y | z | |---|---|---|---|---| | A 30°-60°-90° triangle with a hypotenuse of length *x*, a shorter leg of length y, and a longer leg of length 20. The right angle is labeled *z*. | 40 | 20 | 20√3 | 19. | Diagram | x | y | z | |---|---|---|---|---| | A 30°-60°-90° triangle with a shorter leg of length *x*, a hypotenuse of length *y*, and a longer leg of length 6√3. The right angle is labeled *z*. | 6 | 12 | | 20. | Diagram | x | y | z | |---|---|---|---|---| | A 45°-45°-90° triangle with a leg of length *x*. The hypotenuse is of length 10√6, the right angle is labeled *z*, and y = x. | 5√6 | 5√6 | - | 21. Find the perimeter of the triangle: - A 30°-60°-90° triangle with a shorter leg of length 4√15, a longer leg of length x, and a hypotenuse of length y. 22. Find the perimeter of the square: - A square with side length of 28. 23. Steel loading ramps are used to load a lawn mower onto a truckbed 37.5 inches above ground. If the ramps make a 30° angle with the ground, find the length of the ramps in feet. 24. The infield of a baseball field is a square with sides measuring 90 feet. A ball thrown from third to first base is caught in 1.2 seconds. Find the speed of the ball in feet per second. Round to the nearest tenth. # **Unit 7: Right Triangles & Trigonometry** ## Homework 3: Similar Right Triangles & Geometric Mean #### **Directions:** Identify the similar triangles in the diagram, then sketch them so the corresponding sides and angles have the same orientation. 1. A right triangle *LMK* with altitude *LJ* drawn from the right angle *L* to the hypotenuse *KM*. - Triangle *LMK* is similar to triangle *LJM* and triangle *JLK*. - Triangle *JLK* has sides of length *JL* and *LK* and hypotenuse *JK*. - Triangle *LJM* has sides of length *LJ* and *JM* and hypotenuse *LM*. - Triangle *LMK* has sides of length *LM* and *LK* and hypotenuse *MK*. 2. A right triangle *XYZ* with altitude *WX* drawn from the right angle *X* to the hypotenuse *YZ*. - Triangle *XYZ* is similar to triangle *WXZ* and triangle *WXY*. - Triangle *WXY* has sides of length *WX* and *WY* and hypotenuse *XY*. - Triangle *WXZ* has sides of length *WX* and *WZ* and hypotenuse *XZ*. - Triangle *XYZ* has sides of length *XY* and *XZ* and hypotenuse *YZ*. #### **Directions:** Solve for *x*. 3. A right triangle with legs of lengths 6 and 8, and a hypotenuse of length 10. The altitude is drawn to the hypotenuse and is of length *x*. 4. A right triangle with legs of lengths 20 and 21, and a hypotenuse of length 29. The altitude is drawn to the hypotenuse and is of length *x*. 5. A right triangle with legs of lengths 20 and 48, and a hypotenuse of length 52. The altitude is drawn to the hypotenuse and is of length *x*. 6. A right triangle with legs of lengths 13.2 and 26, and a hypotenuse of length 22.4. The altitude is drawn to the hypotenuse and is of length *x*. #### **Directions:** Find the geometric mean of each pair of numbers. 7. 16 and 27 8. 5 and 36 9. 24 and 32 10. 8 and 48 #### **Directions:** Solve for *x*. | Example | Diagram | x | |---|---|---| | 11. | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 40. The two segments of the hypotenuse are of lengths 12 and 28. The altitude is of length *x*. | 11.3 | | 12. | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 39. The two segments of the hypotenuse are of lengths 3 and 36. The altitude is of length *x*. | 9.5 | | 13. | A right triangle with altitude drawn to the hypotenuse, the longer segment of the hypotenuse is of length 14. One of the legs is of length 4, and the altitude is of length *x*. | 5.3 | | 14. | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 63. The two segments of the hypotenuse are of lengths 55 and 8. The altitude is of length *x*. | 6.6 | | 15. | A right triangle with altitude drawn to the hypotenuse, the longer segment of the hypotenuse is of length 28. One of the legs is of length 18, and the altitude is of length *x*. | 12.7 | | 16. | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 20. The two segments of the hypotenuse are of lengths 8 and 12. The altitude is of length *x*. | 9.8 | | 17. | A right triangle with altitude drawn to the hypotenuse, the longer segment of the hypotenuse is of length 12. One of the legs is of length 16, and the altitude is of length *x*. | 15.5 | | 18. | A right triangle with altitude drawn to the hypotenuse, the longer segment of the hypotenuse is of length 21. One of the legs is of length 9, and the altitude is of length *x*. | 13.4 | #### **Directions:** Solve for *x*, *y*, and *z*. 19. | Diagram | x | y | z | |---|---|---|---|---| | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 15. The two segments of the hypotenuse are of lengths x and y. The altitude is of length *z*. | 6 | 9 | 7.3 | | 20. | A right triangle with altitude drawn to the hypotenuse, hypotenuse of length 30. The two segments of the hypotenuse are of lengths x and y. The altitude is of length *z*. | 12 | 18 | 16.4 | # Unit 7: Right Triangles & Trigonometry ## Homework 4: Trigonometric Ratios & Finding Missing Sides #### **Directions:** Give each trig ratio as a fraction in simplest form. 1. A right triangle with a leg of length 14, a leg of length 50, and a hypotenuse of length 52. The longer leg is opposite angle *Q* and angle *R* is the right angle. - sin Q = 25/26 - sin R = 1 - cos Q = 7/26 - cos R = o - tan Q = 25/7 - tan R = undefined #### **Directions:** Solve for x. Round to the nearest tenth. 2. | Diagram | x | |---|---|---| | A right triangle with a leg of length 17, a leg of length x, and a hypotenuse of length 48. The angle opposite *x* is of measure 62°. | 43.2 | 3. | Diagram | x | |---|---|---| | A right triangle with a leg of length 11, a hypotenuse of length 29, and a leg of length *x*. The angle opposite *x* is of measure 67°. | 27.1 | 4. | Diagram | x | |---|---|---| | A right triangle with a hypotenuse of length 12, a leg of length *x*, and a leg of length 32. The angle opposite *x* is of measure 29°. | 11.3 | 5. | Diagram | x | |---|---|---| | A right triangle with a leg of length 16, a leg of length x, and a hypotenuse of length 37. The angle opposite *x* is of measure 16°. | 6.2 | 6. | Diagram | x | |---|---|---| | A right triangle with a leg of length 22, a leg of length x, and a hypotenuse of length 58. The angle opposite *x* is of measure 58°. | 52.2 | 7. | Diagram | x | |---|---|---| | A right triangle with a leg of length 15, a hypotenuse of length *x*, and a leg of length. The angle opposite *x* is of measure 51°. | 19.5 | 8. | Diagram | x | |---|---|---| | A right triangle with a leg of length 37, a hypotenuse of length 48, and a leg of length *x*. | 29.3 | 9. | Diagram | x| |---|---|---| | A right triangle with a leg of length 9 and a hypotenuse of length *x*. The angle opposite *x* is of measure 24°. | 22.1 | 10.| Diagram | x | |---|---|---| | A right triangle with a leg of length 42, a leg of length x, and a hypotenuse of length 40. The angle opposite *x* is of measure 64°.| 17.5 | 11. | Diagram | x | |---|---|---| | A right triangle with a hypotenuse of length *x*, a leg of length 13, and a leg of length. The angle opposite *x* is of measure 70°. | 13.8 | 12. Find DC: - A right triangle with a hypotenuse of length 28. One of the legs is of length 20, the adjacent acute