Geometry Worksheet Name______________________________________ Inequalities in One Triangle Date_______________________Period__________ In exercises 1-6, the lengths of two sides of a triangle are given. (a) If x is the length of the third side of the triangle and the domain of x is all real numbers, find all possible values for x. (b) If x is the length of the third side of the triangle and the domain of x is the set { 1

2 , 1, 4, 7, 9.3, 14, 19}, find all possible values for x. 1. 10, 10 3. 3.9, 2.3

2. 6 13

, 4 13

4. 2 12

, 1162

5. A Corellian Engineering Corporation YT-1300 stock light freighter travels from Anchorhead, Tatooine to Mos Eisley. From Mos Eisley, it turns 58o and flies to Mos Espa, and then turns 65o to go back to its starting point in Anchorhead. Which of the three legs of the trip is the longest? 6. Three Death Star Turbolaser Towers are positioned to form a triangle. The measure of the angle at tower M is 10o less than the measure of the angle at tower K. The measure of the angle at tower L is 1o greater than the measure of the angle at tower K. Which two towers are closest together? In exercises 7-10, state whether it is possible for a triangle to exist with sides of the given lengths. 7. 3x, 2x, 4x (x>0) 9. x, y, x+y (x and y > 0) 8. x, x+3, x+4 (x > 1) 10. 2x, 2x+2, 4x+3 (x>0) 11. The distance from Coruscant to Yavin is 171 mega light years. The distance from Yavin to Nar Shadda is 206 mega light years. Use the Triangle Inequality Theorem to find the range for the distance from Coruscant to Nar Shadda. 12. In Aurra Sing’s quarters, the distance from the blast door D to the ray shielded armory C is 4 feet. The distance from the armory to the transparisteel window W is 6 feet. On a floor plan of her room, angle CDW is drawn. The distance from the window to the door is 8 feet. Order the angles from least to greatest measure. 13. If AC > CX, and BX > BD, then 14. If m∠VRT = 120 and m∠S > m∠T, then 18. Which side is the longest? which angle is larger, ∠1 or ∠4? the longest side of ΔRST is ________. the shortest?

60

DC

68 48BA

A B

C D

X

12 3

4V R

T

S 15. In ΔABC, if AB ≅BC , 16. Which side is the longest? 17. Which side is the longest? m∠1 = 6x + 2, and m∠2 = 8x – 20, find m∠B. the shortest? the shortest?

I

G Ha

110

a

D

EF

115

A

B

C

1

2

Worksheet Name_____________________________________ Inequalities in 2 Triangles Date______________________Period___________ Hinge Theorem: If 2 sides of a ∆ are ≅ to 2 sides of another triangle, and the included angle of the 1st ∆ is greater than the included angle of the 2nd ∆, then the third side of the 1st ∆ is greater than the third side of the 2nd ∆. I. Complete each statement using sometimes, always, or never.

If m∠S > m∠W, then RT > VX.

A

B

C R S

T

1. If AB ≅ RS and BC ≅ ST , then AC is __________________ ≅ to RT .

2. If AB ≅ RS and AC ≅ RT , then m∠A is ___________________ = to m∠R.

3. If AB ≅ RS , AC ≅ RT , and ∠A ≅ ∠R, then BC is ___________________ ≅ to ST .

4. If AC ≅ RT , BC ≅ ST , and AB > RS, then m∠C is ___________________ > m∠T.

5. If AC ≅ RT , BC ≅ ST , and m∠C > m∠T, then AB is ___________________ > RS.

6. If m∠A = m∠R, m∠B = m∠S, and AB > RS, then AC is ___________________ < RT. II. State whether AB or BD is the greater measure. 7. _______ 8. ________ 9. ________ 10. ______ A

B

C D

1955

37

19

A

B

C

D

20

20

105A B

C

D17

70

96 46

17121

30 30A

B

C

D

III. State whether ∠1 or ∠2 is the larger angle. 11. ______ 12. _______ 13. _______ 14. ______

8 9

161

8 10

162

96

12

60 60

9

12

1 2

1

2

5

4.6 5

4.7

454

11

7

7

1

2

Review: Pythagorean Theorem and more Name: ___________________________ Show your work, and write all answers in simplest exact form. Date: ________________ Period: _____ 1. A right triangle is inscribed in the circle below. Find the radius of the circle.

2. The triangle below has a right angle on the circle. Solve for x and y in the figure below.

______3. Find the length of a side of a square whose diagonal measures 20.

______4. Find the length of a side of an equilateral triangle whose altitude measures 27.

5. Both quadrilaterals below are squares. Find y.

6. Explain why ∆ABC is a right triangle.

7. Use Pythagorean Triples to solve for r.

8. Use Pythagorean Triples to solve for y.

_________9. Find x. _________10. Find x.

11. If JG is the altitude to the base FH of isosceles triangle JFH, FJ = 15, and FH = 24, find JG.

12. A baseball diamond is a square 90 feet on a side. What is the distance from first base to third base (straight across the center)

15 8

x

32 9

r

y

x

12

133

8.5

35

1215 15

x

13. The dimensions of a rectangular doorway are 200 cm by 90 cm. You are going to carry a circular tabletop through the doorway – what is the largest whole number diameter it can have?

14. The size of a television screen is given by the length of the diagonal of the screen. What size is a television screen that is 21.6 inches wide and 16.2 inches high?

15. A rectangle 6 in. wide has a diagonal 10 in. long. Find the perimeter.

16. A man travels 7 mi. due north, 6 mi. due east, and then 4 mi. due north. How far is he from his starting point?

17) In the figure below, QNM ~ ΔPQN, and NM = 12.5 centimeters.

Δ

What is the area of PQN? Δ

18) If XY = 8 feet and XZ = 17 feet, what is the area of ΔXYZ?

19) Which of the following sets of numbers represents the side lengths in units of a right triangle?

A 5, 3.2, 4.1 B 3.6, 6, 4.8 C 4.5, 8, 6.7 D 8.5, 5.2, 10

20) What is the area of the square in the figure?

21) The figure below shows three right triangles joined at their right-angle vertices to form a triangular pyramid. What is the length of XZ ?

22) ΔWXY is isosceles. WY is 10 centimeters long. Find the length of XZ . Which two triangles are congruent? Justify your answer with a Triangle Congruence Postulate. Then explain why

ZYWZ ≅ .

Name:___________________________________ Date________________________Period__________ Special Right Triangles Find the value of a and b for each of the following special right triangles.

60a

4b

30

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

60

a 14

b

60a

10 b

30

a

5 3

b

30

a

4 b

60

12

b

a

45a

6b 2

45

30a

8 b

45

45a

4.5 b

45

60

a

6 b

30

60a

2 3

b

30

30 a

b

60 78

45

a 7

b

605

a b 45 3 2

a b

Name:___________________________________ Date________________________Period__________ For questions 16-21, refer to the diagram at the right.

16. m ADB∠ =

17. 3 2

30 A

BDC =

B18. C =

19. DC 60

AB =

20. AD =

21. AC =

22. At a time of day when the sun can be sighted at an angle of 60 above

the horizon, a flagpole casts a shadow that is 21 ft long. How tall is the flagpole? _____________

23. The perimeter of a square is 72 units. What is the length of the diagonal of the square? _______________

21 ft

60

24. Mr. Schultz has a garden shaped like an equilateral triangle that measures 11 feet on each side. He has placed a watering hose that extends from the faucet located at a vertex to the opposite side, as shown. What is the length of the hose in the garden?

25. In the figure shown, MN = 10 centimeters. What is the length of segment TN in simplified radical form?

Name:___________________________________ Date________________________Period__________

26. Find the perimeter of quadrilateral WXYZ.

27. Find the perimeter of quadrilateral QRST.

28. In bowling, the pins are arranged in a pattern based on equilateral triangles. What is the distance between pins 1 and 5?

29. Suppose ∆JKL is in the first quadrant and that angle K is a right angle. J is located at (2, 7) and K is located at (2, 2). Suppose segment JK is a leg of ∆JKL and that ∆JKL is a 45-45-90 triangle. What are the coordinates of L?

30. Suppose ∆JKL is in the first quadrant and that angle K is a right angle. J is located at (2, 7) and K is located at (2, 2). Suppose segment JK is the side opposite of the 60 degree angle of ∆JKL. What are the coordinates of L in simplified radical form?

Name:___________________________________ Date________________________Period__________

31. A large stained glass window is constructed from six 30-60-90 triangles. What is the height of the window? Assume that the window has reflectional symmetry.

32. Find a, b, c, and d.

33. Find w, x, y, and z.

34. A box of chocolates shaped like a regular hexagon is placed snugly inside of a rectangular box as shown in the figure. If the side length of the hexagon is 3 inches, what are the dimensions of the rectangular box?

35. Kim and Yolanda are watching a movie in a theater. Yolanda is sitting x feet from the screen and Kim is 15 feet behind Yolanda. The angle that Yolanda’s line of sight to the top of the screen makes with the horizontal is 30 degrees. The angle that Yolanda’s line of sight to the top of the screen makes with the horizontal is 45. How far behind the screen are Yolanda and Kim?

Intro to Trigonometry Name________________________________________Date__________________ Period_______

36. A road has a grade of 28.4%. This means that the road rises 28.4 ft over a horizontal distance of 100 ft. What angle does the hill make with a horizontal line? Round to the nearest degree.

37. a) Find the tangent of the acute angle formed by the graph of y = (3/4) x and the x axis.

What is another way of describing the slope of a line?

b) Find the measure of the acute angle formed by the graph of y = (3/4) x and the x axis. Round to the nearest degree.

38. The coordinates of the vertices of ∆RST are R(3, 3), S(8,3) and T(8, -6).

a) What is the measure of angle T to the nearest degree?

b) What is the measure of angle R?

39. If the tangent of angle A is (1/2), what is the 40. If the sine of angle A is (1/2), what is the cosine of angle A in simplified radical form? tangent of angle A in simplified radical form?

Set up the trig ratio for each problem. Then solve for the missing value. 41. 42. 43. 44. x

8

Find any missing lengths to the nearest thousandth. 45. Each base angle of an isosceles 46. f = __________ 47. t = __________ 48. h = __________ triangle measures 50° and the altitude to the base is 26 centimeters long. Find the length of the base.

26°

20

x

24

26

x°

x

26°

20 7

27°

Geometry Name___________________________ WS – Trig Word Problems Date_________ For each of the following problems draw a diagram (if one is not provided), label the diagram, write a trig ratio equation, rewrite the equation so that it is calculator ready and then solve. Round answers to the nearest thousandth.

_______1. A ladder 14 feet long rests against the side of a building. The base of the ladder rests on level ground 2 feet from the side of the building. What angle does the ladder form with the ground?

_______2. A 24-foot ladder leaning against a building forms an 18° angle with the side of the building. How far is the base of the ladder from the base of the building?

_______3. A road rises 35 feet for every 400 horizontal feet. What is the measurement of the angle the road forms with the horizontal?

_______4. A 32-foot ladder leaning against a building touches the side of the building 26 feet above the ground. What is the measurement of the angle formed by the ladder and the ground?

_______5. The directions for the use of a ladder recommend that for maximum safety the ladder should be placed against a wall at a 75° angle with the ground. If the ladder is 14 feet long, how far from the wall should the base of the ladder be placed?

_______6. A kite is held by a taut string pegged to the ground. The string is 40 feet long and makes a 33° with the ground. Supposing that the ground is level, find the vertical distance from the ground to the kite.

14 18° 24 24

x x 2

road 35

400

x

Created by L. English

_______7. A wire anchored to the ground braces a 17-foot pole. The wire is 20 feet long and is attached to the pole 15 feet from the base of the pole. What angle does the wire make with the ground?

_______8. A jet airplane begins a steady climb of 15° and flies for two ground miles. What was its change in altitude?

_______9. A cable is used to support a 245-m television tower. If the angle the cable makes with the ground is 78°, how long is the cable?

_______10. The 600 block of Powell Street in San Francisco has a steep rise. If it takes 66 feet along the horizontal for the street to rise 10 feet, find the angle the street makes with the horizontal.

_______11. A guy wire is attached to a tower at a point 100 feet above the ground. If the tower is perpendicular to the ground and the wire makes an angle of 55° with the ground, find the length of the wire.

_______12. A patient is being treated with radiotherapy for a tumor that is behind a vital organ. In order to prevent damage tot he organ, the radiologist must angle the rays to the tumor. If the tumor is 6.3 cm. below the skim and the rays enter the body 9.8 cm. to the right of the tumor, find the angle the rays should enter the body to hit the tumor.

skin 9.8 cm

x

x 15

2 miles

organ

tumor

15 20

Created by L. English

Geometry Name___________________________ WS – More Trig Word Problems Date_________ Draw a picture, write a trig ratio equation, rewrite the equation so that it is calculator ready and then solve each problem. Round measures of segments to the nearest thousandth and measures of angles to the nearest thousandth.

________1. A 20-foot ladder leans against a wall so that the base of the ladder is 8 feet from the base of the building. What is the ladder’s angle of elevation?

________2. A 50-meter vertical tower is braced with a cable secured at the top of the tower and tied 30 meters from the base. What is the angle of depression from the top of the tower to the point on the ground where the cable is tied?

________3. At a point on the ground 50 feet from the foot of a tree, the angle of elevation to the top of the tree is 53°. Find the height of the tree.

________4. From the top of a lighthouse 210 feet high, the angle of depression of a boat is 27°. Find the distance from the boat to the foot of the lighthouse. The lighthouse was built at sea level.

________5. Richard is flying a kite. The kite string has an angle of elevation of 57°. If Richard is standing 100 feet from the point on the ground directly below the kite, find the length of the kite string.

________6. An airplane rises vertically 1000 feet over a horizontal distance of 5280 feet. What is the angle of elevation of the airplane’s path?

________7. A person at one end of a 230-foot bridge spots the river’s edge directly below the opposite end of the bridge and finds the angle of depression to be 57°. How far below the bridge is the river?

________8. The angle of elevation from a car to a tower is 32°. The tower is 150 ft. tall. How far is the car from the tower?

230

Created by L. English

________9. A radio tower 200 ft. high casts a shadow 75 ft. long. What is the angle of elevation of the sun?

________10. An escalator from the ground floor to the second floor of a department store is 110 ft long and rises 32 ft. vertically. What is the escalator’s angle of elevation?

________11. A rescue team 1000 ft. away from the base of a vertical cliff measures the angle of elevation to the top of the cliff to be 70°. A climber is stranded on a ledge. The angle of elevation from the rescue team to the ledge is 55°. How far is the stranded climber from the top of the cliff? (Hint: Find y and w using trig ratios. Then subtract w from y to find x)

________12. A ladder on a fire truck has its base 8 ft. above the ground. The maximum length of the ladder is 100 ft. If the ladder’s greatest angle of elevation possible is 70°, what is the highest above the ground that it can reach?

________13. A person in an apartment building sights the top and bottom of an office building 500 ft. away. The angle of elevation for the top of the office building is 23° and the angle of depression for the base of the building is 50°. How tall is the office building?

________14. Electronic instruments on a treasure-hunting ship detect a large object on the sea floor. The angle of depression is 29°, and the instruments indicate that the direct-line distance between the ship and the object is about 1400 ft. About how far below the surface of the water is the object, and how far must the ship travel to be directly over it?

110 32

100

x

1000

w y

8

500 1400

Created by L. English

WORKSHEET ON RIGHT TRIANGLES (Special Right ∆s & Pyth Thm) Name: ____________________

2. Find x=_____ y=_____ k=_____ m=_____

12. PM is an altitude of equilateral triangle PKO. If PK = 4, find PM.

x y

k m

6

30 45

_____, _____5. A side of a rhombus has length 8, and the measure of one angle of the rhombus is 60. Find the lengths of the diagonals of the rhombus.

6. MR = 6 MP = ______ PR = ______ PQ = ______

7. BC = 8 3 AC = ______ AB = ______ CD = ______

________8. Find the perimeter of this isosceles trapezoid.

13. Nadia skips 3 m. north, 2 m. east, 4 m. north, 13 m. east, and 1 m. north. How far is Nadia from where she started?

14. Find CD in trapezoid ABCD with bases AD and BC . Find the perimeter of this trapezoid.

17. A man travels 7 mi. due north, 3 mi. due east, and then 3 mi. due south. How far is he from his starting point?

18. The legs of an isosceles triangle are 6 in. long. If the base is 8 in. long, find the length of the altitude to the base.

19. Find the length of an altitude of an equilateral triangle with a side 2 3 in. long.

20. An isosceles right triangle has a 6 in. hypotenuse. Find the length of a leg.

R

Q P M135 30

A

BC

D

30

10

1660

D 30°

B 10

8

C

Review for Pre-AP Test 3.1: Inequalities in ∆s and Right ∆s Name: ___________________________ Date: ________________ Period: _____ In addition to this review, study your homework and quizzes for this unit! S

T W

5X+3

102 6X-2

6. Which side is the longest? the shortest?

A B

C

D6

6

7

71

234

7. In the plane figure, AB = 10, BC =9, CD = 11. Compare the measures of angles 1, 2, and 3. 8. In the plane figure, m∠1 = 100, m∠2 = 90, m∠3 = 102. Compare the lengths of AB, BC, CD, and DA. 9. Given: m∠B > m∠A A

B

C

D

E

m∠E > m∠D Which segment is longer, AD or BE ? 10. Name the longest segment.

a) ________ b) ________

11. Which segment is the longest in the figure? 12. Given: XT < CX; ∠1 ≅ ∠2; ∠3 ≅ ∠4 Which is longer, AT or AC ?

13. Which is shorter, RT or ST ? 14. Write the inequality that describes the range of possible values for x.

W

X

Y

Z

5030

10080

45

55

W

X

Y

Z

40120

20

60

50

70

C B

A D

100°

45° 60° 50°

35° 70° A

X

2 3 1 4

C T

12

12 U

R

S

T 92°

52°

75°

(6x+9)°

111°15

8

12

8

Name _________________________________________ Date _______________________ Period ____ Trig Application Word Problems

1. A ladder 10.4 meters long makes an angle of 68° with the ground as it leans against a building. How far up the building does the ladder reach?

2. Find to the nearest degree the angles of a 3-4-5 triangle.

3. The Eiffel Tower in Paris casts a shadow 150 meters long when the angle of elevation of the sun is 63°. Find the height of the Tower.

4. The angle of depression of the closest point on the ground that is visible over the nose of an airplane is called the cockpit cutoff angle. For a certain plane flying level at an altitude of 620 meters, the cockpit cutoff angle is 13°. Find the line-of-sight distance from the pilot to the closest visible point on the ground.

5. Find the measures of the angles of a rhombus with diagonals 6 and 4. Draw a sketch.

6. What angle has a sin of 8/13?

7. Steve’s old car cannot make it up a hill that rises at an angle of more than 10°. Can he get to the upper level of the public garage if he must drive up the 36.4 meter long driveway to get to the upper level that is 5.7 meters above the ground? Draw a sketch and support your answer.

8. For the safety of people in wheelchairs, the building code in Retireville requires that every convalescent home have a ramp rising at less than 3°. The ramp in the Wicker Home rises 2.1 meters for every 22 meters along the ground. Does it meet the requirement of the building code? Draw a sketch and support your answer.

Bonus: From the observation area of Sears Tower in Chicago, overlooking Lake Michigan, you can see two sailboats going due east from the tower. The angles of depression to the two boats are 42 degrees and 29 degrees. If the observation deck is 1,353 feet high, how far apart are the two boats?

Right Triangle Practice Name: ___________________________ (Test Review!) Date: ________________ Period: _____ 1. Find the length of the altitude of an isosceles trapezoid whose bases are 12 and 18 and whose legs have length 9.

21. The circumference of circle J is .14π What is the value of x? (The circumference of a circle = 2π r)

3. A ladder is 25 feet long and its base is placed 7

feet from a wall. How far up the wall does the ladder reach?

4. A baseball diamond is a square 90 feet on a side. What is the distance from home plate to second base?

5. A wire 30 feet long was fastened to the top of a telegraph pole and made secure to a stake in the ground 24 feet from the foot of the pole. Find the height of the pole.

6. A piece broke off rectangle ABDF, leaving trapezoid ACDF. If BD = 16, BC = 7 FD = 24, and E is the midpoint of FD , what is the perimeter of ΔACE?

7. Find the perimeter of an isosceles right triangle with a 6 cm. hypotenuse.

8. The perimeters of two 30-60-90 triangles are in the ratio of 1:2. If the length of the hypotenuse of the larger triangle is 20 cm., find the length of the longer leg of the smaller triangle.

9. Find x, y, and z.

10. Is it possible to make an umbrella which is 27 inc. long lie flat on the bottom of a suitcase whose dimensions are 24 in. x 10 in. x 4 in.?

B A

C

D F E

60° 2 3

12

z

y

x

11. In ΔABC, angle B is a right angle. AB:BC = 2:3. If AC = 13, find AB and BC.

22. The area of semicircle D is π18 . What is the perimeter of ∆ABC? (The area of a circle = π r2)

13. The measures of the angles of a triangle are in

the ratio 1:2:3. The length of the shortest side of the triangle is 7. Find the length of the longest side.

14. Find lengths x, y, and z.

15. Find the perimeter of hexagon RSTUVW.

23. Using slopes, determine whether the three points (2, 7), (3, 6), and (-4, -1) form a right triangle.

24. A wooden pole was broken during a windstorm. Before it broke, the total height of the pole above the ground was 25 feet. After it broke, the top of the pole touched the ground 15 feet from the base. How tall was the part of the pole that was left standing?

25. A rectangular prism has dimensions 3 by 4 by 12. What is the length of the diagonal across the rectangular prism?

26. The area of right triangle ΔABD is 18. The area of triangle ΔABC is 10. If AB = 4, what is CD?

60°

x

y

z 6

45°

A

B C D

30°

S

R

8 T

6

45° U W

V