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Algebra and Trigonometry

10.2 Non-right Triangles: Law of Cosines

Algebra and Trigonometry10.2 Non-right Triangles: Law of Cosines
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  1. Preface
  2. 1 Prerequisites
    1. Introduction to Prerequisites
    2. 1.1 Real Numbers: Algebra Essentials
    3. 1.2 Exponents and Scientific Notation
    4. 1.3 Radicals and Rational Exponents
    5. 1.4 Polynomials
    6. 1.5 Factoring Polynomials
    7. 1.6 Rational Expressions
    8. Key Terms
    9. Key Equations
    10. Key Concepts
    11. Review Exercises
    12. Practice Test
  3. 2 Equations and Inequalities
    1. Introduction to Equations and Inequalities
    2. 2.1 The Rectangular Coordinate Systems and Graphs
    3. 2.2 Linear Equations in One Variable
    4. 2.3 Models and Applications
    5. 2.4 Complex Numbers
    6. 2.5 Quadratic Equations
    7. 2.6 Other Types of Equations
    8. 2.7 Linear Inequalities and Absolute Value Inequalities
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  4. 3 Functions
    1. Introduction to Functions
    2. 3.1 Functions and Function Notation
    3. 3.2 Domain and Range
    4. 3.3 Rates of Change and Behavior of Graphs
    5. 3.4 Composition of Functions
    6. 3.5 Transformation of Functions
    7. 3.6 Absolute Value Functions
    8. 3.7 Inverse Functions
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  5. 4 Linear Functions
    1. Introduction to Linear Functions
    2. 4.1 Linear Functions
    3. 4.2 Modeling with Linear Functions
    4. 4.3 Fitting Linear Models to Data
    5. Key Terms
    6. Key Concepts
    7. Review Exercises
    8. Practice Test
  6. 5 Polynomial and Rational Functions
    1. Introduction to Polynomial and Rational Functions
    2. 5.1 Quadratic Functions
    3. 5.2 Power Functions and Polynomial Functions
    4. 5.3 Graphs of Polynomial Functions
    5. 5.4 Dividing Polynomials
    6. 5.5 Zeros of Polynomial Functions
    7. 5.6 Rational Functions
    8. 5.7 Inverses and Radical Functions
    9. 5.8 Modeling Using Variation
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  7. 6 Exponential and Logarithmic Functions
    1. Introduction to Exponential and Logarithmic Functions
    2. 6.1 Exponential Functions
    3. 6.2 Graphs of Exponential Functions
    4. 6.3 Logarithmic Functions
    5. 6.4 Graphs of Logarithmic Functions
    6. 6.5 Logarithmic Properties
    7. 6.6 Exponential and Logarithmic Equations
    8. 6.7 Exponential and Logarithmic Models
    9. 6.8 Fitting Exponential Models to Data
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  8. 7 The Unit Circle: Sine and Cosine Functions
    1. Introduction to The Unit Circle: Sine and Cosine Functions
    2. 7.1 Angles
    3. 7.2 Right Triangle Trigonometry
    4. 7.3 Unit Circle
    5. 7.4 The Other Trigonometric Functions
    6. Key Terms
    7. Key Equations
    8. Key Concepts
    9. Review Exercises
    10. Practice Test
  9. 8 Periodic Functions
    1. Introduction to Periodic Functions
    2. 8.1 Graphs of the Sine and Cosine Functions
    3. 8.2 Graphs of the Other Trigonometric Functions
    4. 8.3 Inverse Trigonometric Functions
    5. Key Terms
    6. Key Equations
    7. Key Concepts
    8. Review Exercises
    9. Practice Test
  10. 9 Trigonometric Identities and Equations
    1. Introduction to Trigonometric Identities and Equations
    2. 9.1 Solving Trigonometric Equations with Identities
    3. 9.2 Sum and Difference Identities
    4. 9.3 Double-Angle, Half-Angle, and Reduction Formulas
    5. 9.4 Sum-to-Product and Product-to-Sum Formulas
    6. 9.5 Solving Trigonometric Equations
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  11. 10 Further Applications of Trigonometry
    1. Introduction to Further Applications of Trigonometry
    2. 10.1 Non-right Triangles: Law of Sines
    3. 10.2 Non-right Triangles: Law of Cosines
    4. 10.3 Polar Coordinates
    5. 10.4 Polar Coordinates: Graphs
    6. 10.5 Polar Form of Complex Numbers
    7. 10.6 Parametric Equations
    8. 10.7 Parametric Equations: Graphs
    9. 10.8 Vectors
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  12. 11 Systems of Equations and Inequalities
    1. Introduction to Systems of Equations and Inequalities
    2. 11.1 Systems of Linear Equations: Two Variables
    3. 11.2 Systems of Linear Equations: Three Variables
    4. 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables
    5. 11.4 Partial Fractions
    6. 11.5 Matrices and Matrix Operations
    7. 11.6 Solving Systems with Gaussian Elimination
    8. 11.7 Solving Systems with Inverses
    9. 11.8 Solving Systems with Cramer's Rule
    10. Key Terms
    11. Key Equations
    12. Key Concepts
    13. Review Exercises
    14. Practice Test
  13. 12 Analytic Geometry
    1. Introduction to Analytic Geometry
    2. 12.1 The Ellipse
    3. 12.2 The Hyperbola
    4. 12.3 The Parabola
    5. 12.4 Rotation of Axes
    6. 12.5 Conic Sections in Polar Coordinates
    7. Key Terms
    8. Key Equations
    9. Key Concepts
    10. Review Exercises
    11. Practice Test
  14. 13 Sequences, Probability, and Counting Theory
    1. Introduction to Sequences, Probability and Counting Theory
    2. 13.1 Sequences and Their Notations
    3. 13.2 Arithmetic Sequences
    4. 13.3 Geometric Sequences
    5. 13.4 Series and Their Notations
    6. 13.5 Counting Principles
    7. 13.6 Binomial Theorem
    8. 13.7 Probability
    9. Key Terms
    10. Key Equations
    11. Key Concepts
    12. Review Exercises
    13. Practice Test
  15. A | Proofs, Identities, and Toolkit Functions
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
  17. Index

Learning Objectives

In this section, you will:
  • Use the Law of Cosines to solve oblique triangles.
  • Solve applied problems using the Law of Cosines.
  • Use Heron’s formula to find the area of a triangle.

Suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles as shown in Figure 1. How far from port is the boat?

A triangle whose vertices are the boat, the port, and the turning point of the boat. The side between the port and the turning point is 10 mi, and the side between the turning point and the boat is 8 miles. The side between the port and the turning point is extended in a straight dotted line. The angle between the dotted line and the 8 mile side is 20 degrees.
Figure 1

Unfortunately, while the Law of Sines enables us to address many non-right triangle cases, it does not help us with triangles where the known angle is between two known sides, a SAS (side-angle-side) triangle, or when all three sides are known, but no angles are known, a SSS (side-side-side) triangle. In this section, we will investigate another tool for solving oblique triangles described by these last two cases.

Using the Law of Cosines to Solve Oblique Triangles

The tool we need to solve the problem of the boat’s distance from the port is the Law of Cosines, which defines the relationship among angle measurements and side lengths in oblique triangles. Three formulas make up the Law of Cosines. At first glance, the formulas may appear complicated because they include many variables. However, once the pattern is understood, the Law of Cosines is easier to work with than most formulas at this mathematical level.

Understanding how the Law of Cosines is derived will be helpful in using the formulas. The derivation begins with the Generalized Pythagorean Theorem, which is an extension of the Pythagorean Theorem to non-right triangles. Here is how it works: An arbitrary non-right triangle ABC ABCis placed in the coordinate plane with vertex A Aat the origin, side c cdrawn along the x-axis, and vertex C Clocated at some point ( x,y ) ( x,y )in the plane, as illustrated in Figure 2. Generally, triangles exist anywhere in the plane, but for this explanation we will place the triangle as noted.

A triangle A B C plotted in quadrant 1 of the x,y plane. Angle A is theta degrees with opposite side a, angles B and C, with opposite sides b and c respectively, are unknown. Vertex A is located at the origin (0,0), vertex B is located at some point (x-c, 0) along the x-axis, and point C is located at some point in quadrant 1 at the point (b times the cos of theta, b times the sin of theta).
Figure 2

We can drop a perpendicular from C Cto the x-axis (this is the altitude or height). Recalling the basic trigonometric identities, we know that

cosθ= x(adjacent) b(hypotenuse)  and sinθ= y(opposite) b(hypotenuse) cosθ= x(adjacent) b(hypotenuse)  and sinθ= y(opposite) b(hypotenuse)

In terms of θ, x=bcosθ θ, x=bcosθ and y=bsinθ.  y=bsinθ.  The (x,y) (x,y)point located at C Chas coordinates ( bcosθ,bsinθ ). ( bcosθ,bsinθ ).Using the side ( xc ) ( xc )as one leg of a right triangle and y yas the second leg, we can find the length of hypotenuse a ausing the Pythagorean Theorem. Thus,

  a 2 = (xc) 2 + y 2        = (bcosθc) 2 + (bsinθ) 2 Substitute (bcosθ) forxand (bsinθ)for y.        =( b 2 cos 2 θ2bccosθ+ c 2 )+ b 2 sin 2 θ Expand the perfect square.        = b 2 cos 2 θ+ b 2 sin 2 θ+ c 2 2bccosθ Group terms noting that  cos 2 θ+ sin 2 θ=1.        = b 2 ( cos 2 θ+ sin 2 θ )+ c 2 2bccosθ Factor out  b 2 .   a 2 = b 2 + c 2 2bccosθ   a 2 = (xc) 2 + y 2        = (bcosθc) 2 + (bsinθ) 2 Substitute (bcosθ) forxand (bsinθ)for y.        =( b 2 cos 2 θ2bccosθ+ c 2 )+ b 2 sin 2 θ Expand the perfect square.        = b 2 cos 2 θ+ b 2 sin 2 θ+ c 2 2bccosθ Group terms noting that  cos 2 θ+ sin 2 θ=1.        = b 2 ( cos 2 θ+ sin 2 θ )+ c 2 2bccosθ Factor out  b 2 .   a 2 = b 2 + c 2 2bccosθ

The formula derived is one of the three equations of the Law of Cosines. The other equations are found in a similar fashion.

Keep in mind that it is always helpful to sketch the triangle when solving for angles or sides. In a real-world scenario, try to draw a diagram of the situation. As more information emerges, the diagram may have to be altered. Make those alterations to the diagram and, in the end, the problem will be easier to solve.

Law of Cosines

The Law of Cosines states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle. For triangles labeled as in Figure 3, with angles α,β, α,β, and γ, γ, and opposite corresponding sides a,b, a,b, and c, c,respectively, the Law of Cosines is given as three equations.

a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ a 2 = b 2 + c 2 2bccosα b 2 = a 2 + c 2 2accosβ c 2 = a 2 + b 2 2abcosγ
A triangle with standard labels: angles alpha, beta, and gamma with opposite sides a, b, and c respectively.
Figure 3

To solve for a missing side measurement, the corresponding opposite angle measure is needed.

When solving for an angle, the corresponding opposite side measure is needed. We can use another version of the Law of Cosines to solve for an angle.

cos α= b 2 + c 2 a 2 2bc cos β= a 2 + c 2 b 2 2ac cos γ= a 2 + b 2 c 2 2ab cos α= b 2 + c 2 a 2 2bc cos β= a 2 + c 2 b 2 2ac cos γ= a 2 + b 2 c 2 2ab

How To

Given two sides and the angle between them (SAS), find the measures of the remaining side and angles of a triangle.

  1. Sketch the triangle. Identify the measures of the known sides and angles. Use variables to represent the measures of the unknown sides and angles.
  2. Apply the Law of Cosines to find the length of the unknown side or angle.
  3. Apply the Law of Sines or Cosines to find the measure of a second angle.
  4. Compute the measure of the remaining angle.

Example 1

Finding the Unknown Side and Angles of a SAS Triangle

Find the unknown side and angles of the triangle in Figure 4.

A triangle with standard labels. Side a = 10, side c = 12, and angle beta = 30 degrees.
Figure 4
Try It #1

Find the missing side and angles of the given triangle: α=30°,b=12,c=24. α=30°,b=12,c=24.

Example 2

Solving for an Angle of a SSS Triangle

Find the angle α αfor the given triangle if side a=20, a=20,side b=25, b=25,and side c=18. c=18.

Analysis

Because the inverse cosine can return any angle between 0 and 180 degrees, there will not be any ambiguous cases using this method.

Try It #2

Given a=5,b=7, a=5,b=7,and c=10, c=10,find the missing angles.

Solving Applied Problems Using the Law of Cosines

Just as the Law of Sines provided the appropriate equations to solve a number of applications, the Law of Cosines is applicable to situations in which the given data fits the cosine models. We may see these in the fields of navigation, surveying, astronomy, and geometry, just to name a few.

Example 3

Using the Law of Cosines to Solve a Communication Problem

On many cell phones with GPS, an approximate location can be given before the GPS signal is received. This is accomplished through a process called triangulation, which works by using the distances from two known points. Suppose there are two cell phone towers within range of a cell phone. The two towers are located 6000 feet apart along a straight highway, running east to west, and the cell phone is north of the highway. Based on the signal delay, it can be determined that the signal is 5,050 feet from the first tower and 2,420 feet from the second tower. Determine the position of the cell phone north and east of the first tower, and determine how far it is from the highway.

Example 4

Calculating Distance Traveled Using a SAS Triangle

Returning to our problem at the beginning of this section, suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles. How far from port is the boat? The diagram is repeated here in Figure 8.

A triangle whose vertices are the boat, the port, and the turning point of the boat. The side between the port and the turning point is 10 mi, and the side between the turning point and the boat is 8 miles. The side between the port and the turning point is extended in a straight dotted line. The angle between the dotted line and the 8 mile side is 20 degrees.
Figure 8

Using Heron’s Formula to Find the Area of a Triangle

We already learned how to find the area of an oblique triangle when we know two sides and an angle. We also know the formula to find the area of a triangle using the base and the height. When we know the three sides, however, we can use Heron’s formula instead of finding the height. Heron of Alexandria was a geometer who lived during the first century A.D. He discovered a formula for finding the area of oblique triangles when three sides are known.

Heron’s Formula

Heron’s formula finds the area of oblique triangles in which sides a,b, a,b, and c care known.

Area= s( sa )( sb )( sc ) Area= s( sa )( sb )( sc )

where s= ( a+b+c ) 2 s= ( a+b+c ) 2 is one half of the perimeter of the triangle, sometimes called the semi-perimeter.

Example 5

Using Heron’s Formula to Find the Area of a Given Triangle

Find the area of the triangle in Figure 9 using Heron’s formula.

A triangle with angles A, B, and C and opposite sides a, b, and c, respectively. Side a = 10, side b - 15, and side c = 7.
Figure 9
Try It #3

Use Heron’s formula to find the area of a triangle with sides of lengths a=29.7ft,b=42.3ft, a=29.7ft,b=42.3ft, and c=38.4ft. c=38.4ft.

Example 6

Applying Heron’s Formula to a Real-World Problem

A Chicago city developer wants to construct a building consisting of artist’s lofts on a triangular lot bordered by Rush Street, Wabash Avenue, and Pearson Street. The frontage along Rush Street is approximately 62.4 meters, along Wabash Avenue it is approximately 43.5 meters, and along Pearson Street it is approximately 34.1 meters. How many square meters are available to the developer? See Figure 10 for a view of the city property.

A triangle formed by sides Rush Street, N. Wabash Ave, and E. Pearson Street with lengths 62.4, 43.5, and 34.1, respectively.
Figure 10
Try It #4

Find the area of a triangle given a=4.38ft,b=3.79ft, a=4.38ft,b=3.79ft,and c=5.22ft. c=5.22ft.

Media

Access these online resources for additional instruction and practice with the Law of Cosines.

10.2 Section Exercises

Verbal

1.

If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines?

2.

If you are looking for a missing angle of a triangle, what do you need to know when using the Law of Cosines?

3.

Explain what s srepresents in Heron’s formula.

4.

Explain the relationship between the Pythagorean Theorem and the Law of Cosines.

5.

When must you use the Law of Cosines instead of the Pythagorean Theorem?

Algebraic

For the following exercises, assume α αis opposite side a,β a,β is opposite side b, b,and γ γ is opposite side c. c.If possible, solve each triangle for the unknown side. Round to the nearest tenth.

6.

γ=41.2°,a=2.49,b=3.13 γ=41.2°,a=2.49,b=3.13

7.

α=120°,b=6,c=7 α=120°,b=6,c=7

8.

β=58.7°,a=10.6,c=15.7 β=58.7°,a=10.6,c=15.7

9.

γ=115°,a=18,b=23 γ=115°,a=18,b=23

10.

α=119°,a=26,b=14 α=119°,a=26,b=14

11.

γ=113°,b=10,c=32 γ=113°,b=10,c=32

12.

β=67°,a=49,b=38 β=67°,a=49,b=38

13.

α=43.1°,a=184.2,b=242.8 α=43.1°,a=184.2,b=242.8

14.

α=36.6°,a=186.2,b=242.2 α=36.6°,a=186.2,b=242.2

15.

β= 50° ,a=105,b=45 β= 50° ,a=105,b=45

For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

16.

a=42,b=19,c=30; a=42,b=19,c=30;find angle A. A.

17.

a=14, b=13, c=20; a=14, b=13, c=20; find angle C. C.

18.

a=16,b=31,c=20; a=16,b=31,c=20;find angle B. B.

19.

a=13,b=22,c=28; a=13,b=22,c=28;find angle A. A.

20.

a=108,b=132,c=160; a=108,b=132,c=160;find angle C. C.

For the following exercises, solve the triangle. Round to the nearest tenth.

21.

A=35°,b=8,c=11 A=35°,b=8,c=11

22.

B=88°,a=4.4,c=5.2 B=88°,a=4.4,c=5.2

23.

C=121°,a=21,b=37 C=121°,a=21,b=37

24.

a=13,b=11,c=15 a=13,b=11,c=15

25.

a=3.1,b=3.5,c=5 a=3.1,b=3.5,c=5

26.

a=51,b=25,c=29 a=51,b=25,c=29

For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

27.

Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

28.

Find the area of a triangle with sides of length 20 cm, 26 cm, and 37 cm. Round to the nearest tenth.

29.

a= 1 2 m,b= 1 3 m,c= 1 4 m a= 1 2 m,b= 1 3 m,c= 1 4 m

30.

a=12.4 ft, b=13.7 ft, c=20.2 ft a=12.4 ft, b=13.7 ft, c=20.2 ft

31.

a=1.6 yd, b=2.6 yd, c=4.1 yd a=1.6 yd, b=2.6 yd, c=4.1 yd

Graphical

For the following exercises, find the length of side x. x. Round to the nearest tenth.

32.
A triangle. One angle is 72 degrees, with opposite side = x. The other two sides are 5 and 6.5.
33.
A triangle. One angle is 42 degrees with opposite side = x. The other two sides are 4.5 and 3.4.
34.
A triangle. One angle is 40 degrees with opposite side = 15. The other two sides are 12 and x.
35.
A triangle. One angle is 65 degrees with opposite side = x. The other two sides are 30 and 23.
36.
A triangle. One angle is 50 degrees with opposite side = x. The other two sides are 225 and 305.
37.
A triangle. One angle is 123 degrees with opposite side = x. The other two sides are 1/5 and 1/3.

For the following exercises, find the measurement of angle A. A.

38.
A triangle. Angle A is opposite a side of length 2.3. The other two sides are 1.5 and 2.5.
39.
A triangle. Angle A is opposite a side of length 125. The other two sides are 115 and 100.
40.
A triangle. Angle A is opposite a side of length 6.8. The other two sides are 4.3 and 8.2.
41.
A triangle. Angle A is opposite a side of length 40.6. The other two sides are 38.7 and 23.3.
42.

Find the measure of each angle in the triangle shown in Figure 11. Round to the nearest tenth.

A triangle A B C. Angle A is opposite a side of length 10, angle B is opposite a side of length 12, and angle C is opposite a side of length 7.
Figure 11

For the following exercises, solve for the unknown side. Round to the nearest tenth.

43.
A triangle. One angle is 60 degrees with opposite side unknown. The other two sides are 20 and 28.
44.
A triangle. One angle is 30 degrees with opposite side unknown. The other two sides are 16 and 10.
45.
A triangle. One angle is 22 degrees with opposite side unknown. The other two sides are 20 and 13.
46.
A triangle. One angle is 88 degrees with opposite side = 9. Another side is 5.

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

47.
A triangle with sides 8, 12, and 17. Angles unknown.
48.
A triangle with sides 50, 22, and 36. Angles unknown.
49.
A triangle with sides 1.9, 2.6, and 4.3. Angles unknown.
50.
A triangle with sides 8.9, 12.5, and 16.2. Angles unknown.
51.
A triangle with sides 1/2, 2/3, and 3/5. Angles unknown.

Extensions

52.

A parallelogram has sides of length 16 units and 10 units. The shorter diagonal is 12 units. Find the measure of the longer diagonal.

53.

The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 feet. Find the length of the shorter diagonal.

54.

The sides of a parallelogram are 28 centimeters and 40 centimeters. The measure of the larger angle is 100°. Find the length of the shorter diagonal.

55.

A regular octagon is inscribed in a circle with a radius of 8 inches. (See Figure 12.) Find the perimeter of the octagon.

An octagon inscribed in a circle.
Figure 12
56.

A regular pentagon is inscribed in a circle of radius 12 cm. (See Figure 13.) Find the perimeter of the pentagon. Round to the nearest tenth of a centimeter.

A pentagon inscribed in a circle.
Figure 13

For the following exercises, suppose that x 2 =25+3660cos( 52 ) x 2 =25+3660cos( 52 )represents the relationship of three sides of a triangle and the cosine of an angle.

57.

Draw the triangle.

58.

Find the length of the third side.

For the following exercises, find the area of the triangle.

59.
A triangle. One angle is 22 degrees with opposite side = 3.4. Another side is 5.3.
60.
A triangle. One angle is 80 degrees with opposite side unknown. The other two sides are 8 and 6.
61.
A triangle. One angle is 18 degrees with opposite side = 12.8. Another side is 18.8.

Real-World Applications

62.

A surveyor has taken the measurements shown in Figure 14. Find the distance across the lake. Round answers to the nearest tenth.

A triangle. One angle is 70 degrees with opposite side unknown, which is the length of the lake. The other two sides are 800 and 900 feet.
Figure 14
63.

A satellite calculates the distances and angle shown in Figure 15 (not to scale). Find the distance between the two cities. Round answers to the nearest tenth.

Insert figure(table) alt text: A triangle formed by two cities on the ground and a satellite above them. The angle by the satellite is 2.1 degrees with opposite side unknown, which is the distance between the two cities. The lengths of the other sides are 370 and 350 km.
Figure 15
64.

An airplane flies 220 miles with a heading of 40°, and then flies 180 miles with a heading of 170°. How far is the plane from its starting point, and at what heading? Round answers to the nearest tenth.

65.

A 113-foot tower is located on a hill that is inclined 34° to the horizontal, as shown in Figure 16. A guy-wire is to be attached to the top of the tower and anchored at a point 98 feet uphill from the base of the tower. Find the length of wire needed.

Insert figure(table) alt text: Two triangles, one on top of the other. The bottom triangle is the hill inclined 34 degrees to the horizontal. The second is formed by the base of the tower on the incline of the hill, the top of the tower, and the wire anchor point uphill from the tower on the incline. The sides are the tower, the incline of the hill, and the wire. The tower side is 113 feet and the incline side is 98 feet.
Figure 16
66.

Two ships left a port at the same time. One ship traveled at a speed of 18 miles per hour at a heading of 320°. The other ship traveled at a speed of 22 miles per hour at a heading of 194°. Find the distance between the two ships after 10 hours of travel.

67.

The graph in Figure 17 represents two boats departing at the same time from the same dock. The first boat is traveling at 18 miles per hour at a heading of 327° and the second boat is traveling at 4 miles per hour at a heading of 60°. Find the distance between the two boats after 2 hours.

Insert figure(table) alt text: A graph of two rays, which represent the paths of the two boats. Both rays start at the origin. The first goes into the first quadrant at a 60 degree angle at 4 mph. The second goes into the fourth quadrant at a 327 degree angle from the origin. The second travels at 18 mph.
Figure 17
68.

A triangular swimming pool measures 40 feet on one side and 65 feet on another side. These sides form an angle that measures 50°. How long is the third side (to the nearest tenth)?

69.

A pilot flies in a straight path for 1 hour 30 min. She then makes a course correction, heading 10° to the right of her original course, and flies 2 hours in the new direction. If she maintains a constant speed of 680 miles per hour, how far is she from her starting position?

70.

Los Angeles is 1,744 miles from Chicago, Chicago is 714 miles from New York, and New York is 2,451 miles from Los Angeles. Draw a triangle connecting these three cities, and find the angles in the triangle.

71.

Philadelphia is 140 miles from Washington, D.C., Washington, D.C. is 442 miles from Boston, and Boston is 315 miles from Philadelphia. Draw a triangle connecting these three cities and find the angles in the triangle.

72.

Two planes leave the same airport at the same time. One flies at 20° east of north at 500 miles per hour. The second flies at 30° east of south at 600 miles per hour. How far apart are the planes after 2 hours?

73.

Two airplanes take off in different directions. One travels 300 mph due west and the other travels 25° north of west at 420 mph. After 90 minutes, how far apart are they, assuming they are flying at the same altitude?

74.

A parallelogram has sides of length 15.4 units and 9.8 units. Its area is 72.9 square units. Find the measure of the longer diagonal.

75.

The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is 117°. What is the area of this quadrilateral?

76.

The four sequential sides of a quadrilateral have lengths 5.7 cm, 7.2 cm, 9.4 cm, and 12.8 cm. The angle between the two smallest sides is 106°. What is the area of this quadrilateral?

77.

Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132°. Round to the nearest whole square foot.

78.

Find the area of a triangular piece of land that measures 110 feet on one side and 250 feet on another; the included angle measures 85°. Round to the nearest whole square foot.

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