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ATHS MC – Math Department Al Ain (2013-2014)HOMEWORK TERM 2

GRADE 9 ASP

Name WEEK/1 SCORE: 20

A. Chapter 2.4 : Solving Two- Step and Multi- Step Inequalities.

1. Solve the inequality and graph the solution:

a. −6≥3(x−2) b. 13n+ 1

2< 2

3

c. −2 ( x−7 )−4−x<8 x+32

d. w (2w 1)

2.Write and solve an

a. Two is greater than the sum of 3 and one-fourth of a number.

b. The product of −45 and b is at most -16

.

2 [footer copy]

3. Write and solve an inequality for each problem a. Ann’s grade is the average of four scores: three chapter tests and a final

that counts as two chapter tests. She scored 72, 90, and 75 on the chapter tests. What score does she need on the final to have a grade of at least 80?

b. The average of Maggi’s two test scores must be 80 or more for her to get at least a B in the class. She got a 72 on her first test. What grades can she get on the second test to make at least a B in the class? Check your Answer.

c. Paul is limiting his diet to no more than 600 calories per meal. For lunch, he had a can of iced tea and two bowls of soup. The tea had 140 calories. What are the possible number of calories in each bowl of soup?

NameWEEK/2 08/01

SCORE: 20

B.Chapter 2.5 : Solving Inequalities with Variables on Both sides

1. Solve the inequality and graph the solution:a. 5d+1<−2d−6 b.5 x−1.2<3.1x−2.8

c. 12x−3

4x≥ 1

4 x d. 2x 3x 8

2. Identify each inequality as an identity (all real numbers are solutions) or contradiction (no solutions).

a. 10 2 b.-5(k-1) ≥(2-k) c. 4(x – 1)≤ 4x

3. Write and solve an inequality for each problema. Jay can buy a stereo either online or at a local store. If he buys online, he gets a 15%

discount, but has to pay a $12 shipping fee. At the local store, the stereo is not on sale, but there is no shipping fee. For what regular prices is it cheaper for Jay to buy the stereo online?

b. For what values of x is the area of the rectangle greater than the area of the triangle?


C.Chapter 2.6: Solving Compound Inequalities.

1.Solve each compound inequality and graph the solutions.

1 4x 3 5 b. 3a 5 2 OR 3a 5

c.

p 4 6 AND 3p 18 d. 2 4

2. Write a Compound Inequality shown by each Graph.

a. b.

c d.

3.Write a Compound Inequality for each problem

.a. The United States Postal Service charges a “non machinable surcharge” for first-class mail if the length of the envelope (parallel to the address) divided by the height of the envelope is less than 1.3 or more than 2.5. Charlene has an envelope with a height of 3.5 inches. Write

a compound inequality to show the lengths in inches for which Charlene will have to pay the

surcharge. Graph the solutions.

b. Entertaining: A digital video recorder (DVR) records television shows on an internal hard drive. To use a DVR you need a subscription with a DVR service company. Two companies advertise their charges for a DVR machine and subscription service.

For what numbers of months will a consumer pay less for the machine and subscription at Easy Electronics

than at Cable Solutions ? Write an Inequality and solve .

c. Transportation: The cruise- control function on Peter’s car should keep the speed of the car within 3ml/h of the set speed. Write a Compound Inequality to show the acceptable speeds if the set speed is 55ml/h. Graph the solutions.

4.Write and Graph a Compound Inequality for the numbers described: a. All real Numbers less or equal to 2 and greater than or equal to 1. ………………………………………

b. All real numbers between 10 and -10 inclusive. ………………………………………….,

D.Chapter 2.7 : Solving Absolute Value Inequalities.1. Solve each inequality and graph the solutions

a. x 7 4 b. . x 3 0.7 < 2.7

C. d.

e 6x 2 7 12

f.

2. Write an Absolute Value Equation for each Graph a. b.

c d.

3. Write and solve an Absolute Value Inequality for the following

a. The ideal temperature for a refrigerator is 36.5 °F. It is acceptable for the temperature to differ from this value by at most 1.5 °F. Write and solve an absolute-value inequality to find the range of acceptable temperatures.

b. At a trout farm, most of the trout have a length of 23.5 cm. The length of some of the trout differs from this by as much as 2.1 cm. Write and solve an absolute-value inequality to find the range of lengths of the trout.


E.Chapter 4.3 : Rates of Change and Slope.1. a. The table shows a person’s heart rate over time.Graph the data and show the rate of change.

b. The table shows the distance a car drove on one tank of gasoline.

Miles driven 0 60 150 170 230 260

Gas Used (gal) 0 2 5 6 9 11

a. Graph the data and show the rates of change. b. The rate of change represents the gas mileage in miles

per gallon. Between which two measurements was thecar’s gas mileage least?

2. Find the slope of each line:a. b. c.

3.Tell whether the slope of each line is positive, negative, zero, or undefined.

a. b. c.

F.Chapter 6.1: Properties and Attributes of Polygons.

1. Match each vocabulary term on the left with a part of polygon ABCDE on the right.

1. a diagonal _________ A. point D

2. a side of the polygon _________ B.

3. a vertex of the polygon _________ C.

2. Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.a.. b.. c..

3. Tell whether each polygon is regular or irregular. A concave polygon has a pair of sides that make a “cave” in the polygon. Tell whether each polygon is concave or convex.

a. b.

c.

4. Find the sum of the interior angle measures of a 14-gon. ____________________

5..Find the measure of each interior angle of hexagon ABCDEF.______________________________________________________________________________________________________________________

6 . Do you believe there is an upper limit to the sum of the interior angle measures in n-gons? Explain your reasoning.

7. .Find the value of n in pentagon PQRST.

8. Any regular polygon can be inscribed in a circle. For Exercises a-dfind the length of a side of the regular polygon in terms of r, the radius of the circle. Give the lengths in simplest radical form.

a. regular quadrilateral (square)__________________________________________

b.regular octagon (Hint: The dotted lines show a square.)__________________________________________

c.regular hexagon__________________________________________

d. regular dodecagon (Hint: The dotted lines show a regular hexagon.)__________________________________________

9. Before electric or steam power, a common way to power machinery was with a waterwheel. The simplest form of waterwheel is a series of paddles on a frame partially submerged in a stream. The current in the stream pushes the paddles forward and turns the frame. The power of the turning frame can then be used to drive machinery to saw wood or grind grain. The waterwheel shown has a frame in the shape of a regular octagon.

a. Find the measure of one interior angle of the waterwheel.

b. Find the measure of one exterior angle of the waterwheel.


G. Chapter 6.2: Properties of Parallelogram.

1. Fill in the blanks to complete each definition or theorem.a. If a quadrilateral is a parallelogram, then its consecutive angles are

____________________.

b. If a quadrilateral is a parallelogram, then its opposite sides are ____________________. c. A parallelogram is a quadrilateral with two pairs of ____________________ sides. d. If a quadrilateral is a parallelogram, then its diagonals ____________________

each other. e. If a quadrilateral is a parallelogram, then its opposite angles are ____________________.

2.A gurney is a wheeled cot or stretcher used in hospitals. Many gurneys are made so that the base will fold up for easy storage in an ambulance. When partially folded, the base forms a parallelogram. In

STUV, VU 91 centimeters, UW 108.8 centimeters, and mTSV 57. Find each measure.a.. SW b. TS c.. US

__________________________ __________________________

__________________________

d.. mSVU e.. mSTU f.. mTUV__________________________ __________________________

__________________________

3. The area of a parallelogram is given by the formula Abh, where A is the area, b is the length of a base, and h is the height perpendicular to the base. ABCD is a parallelogram. E, F, G, and H are the midpoints of the sides.

a..Show that the area of EFGH is half the area of ABCD.

b. Show that EFGH is a parallelogram.

4. Mr. Nguyen is blessed (or cursed) with an abundance of books. They litter his apartment. Mr. Nguyen is trying to clean up his living room, so he bought a new three-shelf bookcase. Each

shelf is feet long.

a. Mr. Nguyen measures a few books and finds they average inch thick. Calculate the maximum number of books Mr. Nguyen can fit on the bookcase. _____________

b. . Mr. Nguyen finds that if he stresses a book by pushing back on the front cover, the book gets thinner. Mr. Nguyen discovers that he can stress a book by about 30 without harming the book’s binding. Calculate the maximum number of stressed books Mr. Nguyen can fit on the bookcase. _______________

5. Find the range of possible diagonal lengths in a parallelogram with the given side lengths.

a. 3 and 12 b. x and 2x c. x and x __________________________ __________________________

__________________________

H. Chapter 6.3: Conditions of Parallelogram.

1. For each definition or theorem, tell what information you would need about the figure to conclude need about the figure to conclude that the figure is a parallelogram. For some exercises, there is more than one correct answer, but give only one example per exercise.

a. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. _____________________________

b. If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram. _____________________________

c. If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.

__________________________________________________________________________________________

d. If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.

__________________________________________________________________________________________

e. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (Hint: The diagonals of the figure are and )

__________________________________________________________________________________________

f. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. _____________________________

2. A quadrilateral has vertices E(1, 1), F(4, 5), G(6, 6), H(3, 2). Complete Exercises a-d to tell whether EFGH is a parallelogram.

g. Plot the vertices and draw EFGH. h. Use the Distance Formula: EF ________ HG ________

i. Use the Slope Formula: slope of ________

slope of ________

j.The answers to Exercises h and I prove that EFGH is a parallelogram. Which one of Exercises a-f states the theorem that you used? ________

3 1.In parallelogram ABCD, AC and BD . is contained in the line

with equation y and is contained in the line with equation If A and B are both in Quadrant I, find the vertices of ABCD. (Hint: All coordinates are in integers.)

Use the figure for Exercises 2 and 3. A B and C D. 2. Name the conditions under which the figure would be a parallelogram.

__________________________________________________________________________________________

3. Tell which sides of the figure must be parallel under all conditions. Explain your answer.

4. Sketch a parallelogram with perpendicular diagonals. Tell which sides and angles of the parallelogram have to be congruent. Explain your answer.

5. Sketch a parallelogram with perpendicular congruent diagonals. Tell which sides and angles of the parallelogram have to be congruent. Explain your answer.

6. Find the measure(s) of the parallelogram’s angles in your sketch for Exercise 5. _______


I. Chapter 6.4: Properties of Special Parallelogram.A.Match each figure with the letter of one of the vocabulary terms. Use each term once.

1. 2. 3.

____________________ ____________________ ____________________

Fill in the blanks to complete each theorem.4. If a parallelogram is a rhombus, then its diagonals are ___________________.5. If a parallelogram is a rectangle, then its diagonals are ___________________.6. If a quadrilateral is a rectangle, then it is a ___________________.7. If a parallelogram is a rhombus, then each diagonal ___________________

a pair of opposite angles.8. If a quadrilateral is a rhombus, then it is a ___________________.

The part of a ruler shown is a rectangle

with AB 3 inches and BD inches. Find each length.

9. DC _________________________10. AC _________________________

Use the phrases and theorems from the Word Bank to complete this two-column proof.

11. Given: GHIJ is a rhombus.

Prove: 1 3

Statements Reasons

1. GHIJ is a rhombus. 1. Given

2. a. ________________________________ 2. rhomb.

3. 3. opp. sides i4. 1 2 4. b. ________________________________

5. c. ________________________________ 5. rhomb. each diag. bisects opp. ?6. 1 3 6. d. ________________________________

B. A modern artist’s sculpture has rectangular faces. The face shown here is 9 feet long and 4 feet wide. Find each measure in simplest radical form. (Hint: Use the Pythagorean Theorem.)

4. DC _____________________ 5. AD _____________________

6. DB _____________________ 7. AE _____________________

A. rectangle

B. rhombus

Alternate Interior Thm. GHIJ is a parallelogram. Trans. Prop. of 2 3

VWXY is a rhombus. Find each measure.

8. XY _____________________

9. mYVW _____________________

10. mVYX _____________________

11. mXYZ _____________________

12. The vertices of square JKLM are J(2, 4), K(3, 1), L(2, 2), and M(3, 3). Find each of the following to show that the diagonals of square JKLM are congruent perpendicular bisectors of each other.

JL ________ KM ________

slope of ________ slope of ________

midpoint of (________, ________) midpoint of (________, ________)

Write a paragraph proof.

13. Given: ABCD is a rectangle.

Prove: EDC ECD

C. For Exercises 1–5, give your answers in simplest radical form.

1. Find the length of the diagonals of a rectangle with sides of lengths a and b. ____________________

2. Find the length of the diagonals of a square with sides of length a. ____________________

3. Find the length of the sides of a square with diagonals of length a. ____________________

4. Find the length of the sides of a rhombus with diagonals of lengths a and b. ____________________

5. Find the length of a rectangle with width x and a diagonal of length 2x. ____________________

6. Find the measures of the angles in the triangles formed by one diagonal of the rectangle in Exercise 5. ____________________

The figure shows a kind of quadrilateral called a kite. A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. Use the figure to write paragraph proofs for Exercises 7 and 8.

7. Prove: CBA CDA

8. Prove: is the perpendicular bisector of .

For Exercises 9–11, name all the types of quadrilaterals (kite, parallelogram, rectangle, rhombus, or square) that satisfy the given conditions.

9. The diagonals bisect each other. __________________________________________________________________________________________

10. The diagonals are perpendicular. __________________________________________________________________________________________

11. The diagonals are congruent. __________________________________________________________________________________________

SWQ REVISION AND QUIZName WEEK/7 SCORE: 20

J. Chapter 6.5 Conditions of Special Parallelograms.A.Fill in the blanks to complete each theorem. (Hint: Exercise 3 is not stated as

a theorem in the textbook.)1. If one pair of consecutive sides of a parallelogram are congruent, then the

parallelogram is a _____________________.

2. If the diagonals of a parallelogram are _____________________, then the parallelogram is a rhombus.

3. If a parallelogram is both a _____________________ and a _____________________, then the parallelogram is a square.

4. If the _____________________ of a parallelogram are congruent, then the parallelogram is a rectangle.

5. If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a _____________________.

6. If one angle of a parallelogram is a right angle, then the parallelogram is a _____________________.

B.The Jamaican flag is a quadrilateral with a diagonal gold “X” that divides the flag into two black triangles and

two green triangles. In ABCD, and

The diagonals, and are also congruent. Fill in the blanks in Exercises 7 and 8 to show why the flag is a rectangle.

7. Because both pairs of opposite ____________________ are congruent, the flag is a parallelogram.

8. Because ABCD is a parallelogram and the diagonals are ____________________, the flag is a rectangle.

C. Complete Exercises 9–12 to show that the conclusion is valid.

Given: and M is a right angle.

Conclusion: JKLM is a square.

9. Because and JKLM is a _____________________. 10. Because JKLM is a parallelogram and M is a right angle, JKLM is a

_____________________.

11. Because JKLM is a parallelogram and JKLM is a _____________________. 12. Because JKLM is a _____________________ and a _____________________, JKLM is a square.

D. On the National Mall in Washington, D.C., a reflecting pool lies between the Lincoln

Memorial and the World War II Memorial. The pool has two 2300-foot-long sides and two 150-foot-long sides. Tell what additional information you need to know in order to determine whether the reflecting pool is a rectangle. (Hint: Remember that you have to show it is a parallelogram first.) ________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

E. Use the figure for Exercises 2–5. Determine whether each conclusion is valid. If not, tell what additional information is needed to make it valid.

2. Given: and bisect each other. Conclusion: ABCD is a square. __________________________________________________________________________________________

3. Given: Conclusion: ABCD is a rhombus. __________________________________________________________________________________________

4. Given: mADB mABD 45Conclusion: ABCD is a square.__________________________________________________________________________________________

5. Given: Conclusion: ABCD is a rectangle. __________________________________________________________________________________________

F. Find the lengths and slopes of the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all names that apply.

6. E(2, 4), F(0, 1), G(3, 1), H(5, 2) _____________________________________EG ________ FH ________


7. P(1, 3), Q(2, 5), R(0, 4), S(1, 2) _____________________________________PR ________ QS ________


G.1.Of a parallelogram, rectangle, rhombus, or square, name the two that can have a diagonal congruent to a side. Explain your reasoning. ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

2. Of the two quadrilaterals you named in Exercise 1, with a diagonal congruent to a side, tell which one’s diagonal length can be determined unambiguously given a side length (using only what you have learned in this geometry course so far). ____________________

3. If the congruent side and diagonal of the special quadrilateral from Exercise 2 have length x, find the length of the other diagonal. ____________________

4. Give the measures of the interior angles of the special quadrilateral from Exercise 2. ____________________

5. Given the coordinates of three vertices of a parallelogram, tell how many parallelograms can be formed. ____________________

6. Given the coordinates of three vertices of a rectangle, tell how many rectangles can be formed. ____________________

7. Given the coordinates of three vertices of a rhombus, tell how many rhombuses can be formed. ____________________

8. Given the coordinates of three vertices of a square, tell how many squares can be formed. ____________________

9. Given the coordinates of two vertices of a parallelogram, rectangle, or rhombus, tell how many of each can be formed. ____________________

10. Given the coordinates of two vertices of a square, tell how many squares can be formed. ____________________

11. Tell how many points are sufficient to determine a parallelogram. ____________________12. Tell how many points are sufficient to determine a rectangle,

a rhombus, or a square. ____________________13. Find all the possible coordinates of the fourth vertex of a

parallelogram with vertices (2, 3), (3, 0), and (4, 4). ____________________14. Plot the three vertices given in Exercise 13.

Connect the vertices to make a triangle. Then plot the coordinates of all the possible fourth vertices. Connect these vertices.Name the relationship of the original triangle to the new figure you created.__________________________________________

K. Chapter 6.6 :Properties of Kites and Trapezoids.A. Fill in the blanks to complete each theorem or definition.

1. If a quadrilateral is a kite, then its ________________________ are perpendicular. 2. If a quadrilateral is a kite, then exactly one pair of opposite ________________________

are congruent. 3. A kite is a quadrilateral with exactly two pairs of congruent

consecutive ________________________.

B.ABCD is a kite. Use the figure to find each measure in Exercises 4–6.

4. mD 5. AB 6. CD __________________________ __________________________

__________________________

C.A trapezoid is a quadrilateral with exactly one pair of parallel sides. Name the parts of trapezoid PQRS asked for in Exercises 7–9.

7. both bases ____________________

8. both legs ____________________

9. one pair of base angles ____________________

D.Fill in the blanks to complete each theorem or definition.

10. A trapezoid is isosceles if and only if its ____________________ are congruent.

11. If a trapezoid has one pair of congruent base angles, then the trapezoid

is ____________________.

12. If the legs of a trapezoid are ____________________, then the trapezoid is an isosceles trapezoid.

13. If a quadrilateral is an isosceles trapezoid, then each pair of

____________________ is congruent.

E. In an art museum, a statue sits on a pedestal with sides that are isosceles trapezoids. Name the parts of isosceles trapezoid EFGH asked for in Exercises 14 and 15.

14. both pairs of congruent angles _____________________________________

15. both pairs of congruent segments _____________________________________

F.In kite ABCD, mBAC 35 and mBCD 44. For Exercises 1–3, find each measure.

1. mABD 2. mDCA 3. mABC__________________________ __________________________

__________________________

4. Find the area of EFG. _____________________

5. Find mZ. 6. KM 7.5, and NM 2.6. Find LN.

__________________________________________ __________________________________________

7. Find the value of n so that PQRS is isosceles. __________________________________________

8. Find the value of x so that EFGH is isosceles.__________________________________________

9. BD 7a 0.5, and AC 5a 2.3. Find the value of a so that ABCD is isosceles.

__________________________________________

10. QS 8z2, and RT 6z2 38. Find the value of z so that QRST is isosceles.__________________________________________

G.Use the figure for Exercises 11 and 12. The figure shows a ziggurat. A ziggurat is a stepped, flat-topped pyramid that was used as a temple by ancient peoples of Mesopotamia. The dashed lines show that a ziggurat has sides roughly in the shape of a trapezoid.

11. Each “step” in the ziggurat has equal height. Give the vocabulary term for __________________________________________

12. The bottom of the ziggurat is 27.3 meters long, and the top of the ziggurat is 11.6 meters long. Find MN.

H.Use the figure of kite ABCD for Exercises 1–3. 1. The figure shows kite ABCD. Find a formula for the

area of a kite in terms of the diagonals AC and BD. __________________________________________________________________________________________

2. Suppose you are given BA, AC, and ED. Tell whether it is possible to find the area of ABCD. Explain your answer. ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3. Suppose you are given BA, DA, and BD. Tell whether it is possible to find the area of ABCD (with what you have learned so far in this geometry class). Explain your answer. ____________________________________________________________________________________________________________________________________________________________________________________

I. Use the figure of trapezoid PQRS for Exercise 4. 4. Write a paragraph proof.

Given: Prove: PQUT is a rectangle.

J.Write a paragraph proof.5. Given: Isosceles trapezoid JKLM

Prove: JNM is isosceles.


L. Chapter 8.2: Trigonometric Ratios.A.In Exercises 1–3, fill in the blanks to complete each definition. Then use side lengths from the figure to complete the indicated trigonometric ratios.

1. The sine (sin) of an angle is the ratio of the length of the leg _____________________ the angle to the length of the _____________________.

2. The cosine (cos) of an angle is the ratio of the length of the leg _____________________ to the angle to the length of the _____________________.

3. The tangent (tan) of an angle is the ratio of the length of the leg _____________________ the angle to the length of the leg _____________________ to the angle.

B.Use the figure for Exercises 4–6. Write each trigonometric ratio as a simplified fraction and as a decimal rounded to the nearest hundredth.

4. sin L 5. cos L 6. tan M __________________________ __________________________

__________________________

C.Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.7. sin 33 8. cos 47 9. tan 81

__________________________ __________________________

__________________________

D.Use a calculator and trigonometric ratios to find each length. Round to the nearest hundredth.

10. 11. 12.

BD __________________ QP __________________ RS __________________

13. The glide slope is the path a plane uses while it is landing on a runway. The glide slope usually makes a 3 angle with the ground. A plane is on the glide slope and is 1 mile (5280 feet) from touchdown. Use the tangent ratio and a calculator to find EF, the plane’s altitude, to the nearest foot. ____________________

E.Use the figure for Exercises 1–6. Write each trigonometric ratio as a simplified fraction and as a decimal rounded to the nearest hundredth.

1. sin A 2. cos B 3. tan B__________________________ __________________________

__________________________

4. sin B 5. cos A 6. tan A__________________________ __________________________

__________________________

F.Use special right triangles to write each trigonometric ratio as a simplified fraction.

7. sin 30 ________ 8. cos 30 ________ 9. tan 45 ________

10. tan 30 ________ 11. cos 45 ________ 12. tan 60 ________

G.Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.

13. sin 64° ________ 14. cos 58° ________ 15. tan 15° ________

H.Find each length. Round to the nearest hundredth.

16. 17. 18.

XZ _____________ HI _____________ KM _____________

19. 20. 21.

ST _____________ EF _____________ DE _____________

1.Given the lengths of two sides of a triangle and the measure of the included angle, the area of the triangle can be found. In the figure, suppose the lengths b and c and the measure of A are known. Develop a formula for finding the area. Explain your answer. (Hint: Draw an altitude.)____________________________________________________________________________________________________________________________________________________________________________________

__________________________________________________________________________________________

J.Use the formula you developed in Exercise 1 to find the area of each triangle. Round to the nearest hundredth.

2. 3. 4.

__________________________ __________________________

__________________________

5. The law of cosines is a formula to find the length of the third side of any triangle given the lengths of two sides and the measure of the included angle. (Note: The law of cosines applies to any triangle, but deriving it for an obtuse triangle requires more knowledge than you have learned so far.) In the figure, suppose the lengths b and c

and the measure of A are known. Develop the law of cosines to find a. Explain your answer. (Hint: Use the cosine function and the Pythagorean Theorem.)

Use the formula you developed in Exercise 5 to find the missing side length in each triangle. Round to the nearest hundredth.

6. 7. 8.

__________________________ __________________________

__________________________

M. Extension: Trigonometric Ratios of Complementary Angles.See Classwork.

Name WEEK 9 SCORE: 20

14. Chapter 8.3: Solving Right Triangles.A.In Exercises 1–3, fill in the blanks to complete the description of the inverse trigonometric ratios.

1. If sin A x, then sin1 x ________. 2. If cos A ________, then cos1 x mA. 3. If tan A x, then ________ mA.

B.Use the given trigonometric ratio to determine whether 1 or 2 is A in each exercise.

4. ________ 5. ________ 6. ________

7. ________ 8. ________ 9. ________

C.Use a calculator to find each angle measure to the nearest degree.10. sin1 (0.33) ________ 11. cos1 (0.47) ________ 12. tan1 (1.21) ________

13. ________ 14. ________ 15. ________

D.Use a calculator and inverse trigonometric ratios to find the unknown side lengths and angle measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.16. 17. 18.

AC ________ DE ________ GH ________

mB ________ EF ________ mH ________mC ________ mD ________ mI ________

XYZ has vertices X(6, 6), Y(6, 3), and Z(1, 3). Complete Exercises 19–21 to find the side lengths to the nearest hundredth and the angle measures to the nearest degree.19. Plot the points and draw XYZ.

20. Tell which angle is the right angle. ________21. Find XY and YZ from the graph. Use the

Pythagorean Theorem to find XZ.XY ________ YZ ________ XZ ________

Use the given trigonometric ratio to determine which angle of the triangle is A.

1. ________ 2. ________ 3. ________

4. ________ 5. ________ 6. ________

Use a calculator to find each angle measure to the nearest degree.

7. sin1 (0.82) ________ 8. ________ 9. tan1 (5.03) ________

10. ________ 11. cos1 (0.23) ________ 12. ________

Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

13. 14. 15.

__________________________ __________________________

____________________________________________________ __________________________

__________________________

16. 17. 18.

__________________________ __________________________

____________________________________________________ __________________________

__________________________

For each triangle, find all three side lengths to the nearest hundredth and all three angle measures to the nearest degree.19. B(2, 4), C(3, 3), D(2, 3)

__________________________________________________________________________________________

20. L(1, 6), M(1, 6), N(1, 1) __________________________________________________________________________________________

21. X(4, 5), Y(3, 5), Z(3, 4)

A Pythagorean triple is a set of whole numbers that satisfies the Pythagorean Theorem. Exercises 1–4 show Pythagorean triples. Find the measures of the two acute angles, to the nearest degree, in triangles with sides of these lengths.

1. 3, 4, 5 ___________________ 2. 5, 12, 13 ___________________

3. 8, 15, 17 ___________________ 4. 7, 24, 25 ___________________

5. In the United States, common units used for measuring length are inches, feet, and miles. Most of the rest of the world uses centimeters, meters, and kilometers. Would a 20% grade in America be equal to a 20% grade elsewhere? Explain why or why not.

Use the law of sines to find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

6. 7. 8.

__________________________ __________________________

____________________________________________________ __________________________

__________________________

__________________________ __________________________

__________________________

9. 10. 11.

__________________________ __________________________

____________________________________________________ __________________________

__________________________

15. Chapter 8.4: Angles of Elevation and Depression.In Exercises 1 and 2, fill in the blanks to complete the definitions.

1. An angle of elevation is the angle formed by a _______________ line and a line of sight to a point _______________ the line.

2. An angle of _______________ is the angle formed by a horizontal line and a line of sight to a point _______________ the line.

Ben is on the diving board at the neighborhood pool. Jenna is in the pool, and a lifeguard sits at her station on the opposite end of the pool. Classify each angle as an angle of elevation or an angle of depression.

3. 1 ______________________ 4. 2 ______________________

5. 3 ______________________ 6. 4 ______________________

Lisa sees a bird’s nest high in a tree. She decides to use trigonometry to estimate how high the nest is.

7. Lisa walks 15 feet from the base of the tree. She measures an angle of elevation from the ground to the nest of 62. Find how high the nest is above the ground, to the nearest foot.______________________________

8. Lisa spots the mother bird on a branch above the nest. She measures an angle of elevation to the bird of 67. Find how high the mother bird is above the ground, to the nearest foot. ______________________________

Zelda, a trapeze artist, stands on a 10-meter-high platform.9. Zelda measures a 40 angle of depression to

the base of the other platform. Find the distance between the bases of the platforms to the nearest tenth of a meter______________________________

10. Zelda’s partner, Zev, is on the ground doing a safety check on the net. Zelda measures a 79 angle of depression to Zev. Find the distance to the nearest tenth of a meter from Zev to the base of Zelda’s platform. ______________________________

Marco breeds and trains homing pigeons on the roof of his building. Classify each angle as an angle of elevation or an angle of depression.

1. 1 _____________________

2. 2 _____________________

3. 3 _____________________

4. 4 _____________________

To attract customers to his car dealership, Frank tethers a large red balloon

to the ground. In Exercises 5–7, give answers in feet and inches to the nearest inch. (Note: Assume the cord that attaches to the balloon makes a straight segment.)

5. The sun is directly overhead. The shadow of the balloon falls 14 feet 6 inches from the tether. Frank sights an angle of elevation of 67. Find the height of the balloon. ______________________

6. Find the length of the cord that tethers the balloon. ______________________7. The wind picks up and the angle of elevation changes to 59.

Find the height of the balloon. ______________________

Lindsey shouts down to Pete from her third-story window.8. Lindsey is 9.2 meters up, and the angle of depression from

Lindsey to Pete is 79. Find the distance from Pete to the base of the building to the nearest tenth of a meter.

______________________

9. To see Lindsey better, Pete walks out into the street so he is 4.3 meters from the base of the building. Find the angle of depression from Lindsey to Pete to the nearest degree.

______________________10. Mr. Shea lives in Lindsey’s building. While Pete is still out in

the street, Mr. Shea leans out his window to tell Lindsey and Pete to stop all the shouting. The angle of elevation from Peteto Mr. Shea is 72°. Tell whether Mr. Shea lives above or below Lindsey.

Light moves through air at a constant speed. Laser range finders use this fact to gauge distance. The range finder shoots laser light at a distant object and records the time it takes for the reflection of the light to return. One half the time divided by the speed of light gives the distance to the object. If the angle of elevation or depression is also known, the vertical and horizontal distances can be found. In Exercises 1–3, give answers in feet and inches to the nearest inch.

1. A surveyor sights the top of a building with a handheld range finder. The top of the building is 148 feet 2 inches away. The angle of elevation is 56. Find the distance from the surveyor to the building. __________________

2. The angle of depression to the bottom of the building is 4. Find the height of the surveyor. __________________

3. Find the height of the building. __________________

Lamar is on top of a 15-meter cliff. He knows that it is 11.7 meters from the bottom of the cliff to the near bank of a river and that the angle of depression is 52. The angle of depression to the far bank

is 38. Lamar’s calculator does not have trigonometric function keys. 4. Describe how Lamar can find the width of the river.

(Hint: What is special about the angles of depression?)

5. Find the width of the river to the nearest tenth of a meter.__________________________________________________________________________________________

Charlie is late for the hot-air balloon rally. He stops his car 65 meters from a hot-air balloon already taking flight.

6. While Charlie watches the balloon ascend, he tilts his eyes up by 28 until he is watching at an angle of elevation of 66. Find the vertical distance to the nearest tenth of a meter that the balloon ascended while Charlie watched.__________________________________________________________________________________________

7. Charlie watched the balloon ascend for 2 minutes in Exercise 6. The balloon continues to rise at that same rate, but it is now also caught by a 17.7 km/h

horizontal breeze coming from directly behind Charlie. Find the balloon’s angle of ascent to the nearest degree and speed to the nearest tenth of a km/h.

NameWEEK/10,11,12 08/01

REVISION WORK SHEET EXAM

HOMEWORK TERM 2 Gr 9 ASP9

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