Module VI Angles & Triangles

Transcription

1 Module VI Angles & Triangles The word geometry dates back to the Ancient Greeks and literally means earth measure ; geo meaning earth, and metry or metri, meaning measure. Geometry is used in medical imaging such as CAT scans and sonograms; firefighters use geometry to locate a position in the field as well as finding the slope of an incline; accountants use coordinate geometry to display financial data with graphs; professional chefs use geometry to find volume; early childhood educators teach children the language of geometry as well as symmetry and angles to help them describe designs and patterns; In the practice of graphic design, understanding basic geometry is critical to their success; photographers use geometry to help produce images that are visually pleasing; solar photovoltaic installer s use geometry to help them position solar panels so that the sun's energy is optimized; oil industry technicians use geometry and angles in pipe fitting; and ski areas use geometry to find the slope of an incline and when designing jumps. There are many branches of geometry. In this module, we will be focusing on Euclidean Geometry (geometry that adheres to Euclid s axioms, more specifically, the parallel postulate) with respect to angles and triangles and in Module VII, we will continue our study of Euclidean geometry with circles, polygons, and geometric solids. I. The Language of Angles The following are terms that are frequently used in geometry and angles. 1. Line: A line is determined by two distinct points and extends indefinitely in both directions. Lines in the Euclidean plane are either parallel, or they intersect in one point. Below is the line AB. The notation that is used to describe the line AB is AB. A B 2. Ray: A ray starts at a point and extends indefinitely in one direction. Below is the ray AB. The notation that is used to describe the ray AB is AB. A B MAT 107 Career Math Angles & Triangles 149

2 3. Line Segment: A line segment is a part of a line and has two endpoints. Below is the line segment AB. The notation that is used to describe the line segment AB is AB. A B 4. Perpendicular Lines: Perpendicular lines intersect to form right angles. The box in the corner indicates that the two lines form a right angle. The notation that is used to describe two perpendicular lines l 1 (line 1) and l 2 (line 2) is l1 l2. l 1 l 2 5. Parallel Lines: Lines that are parallel in Euclidean Geometry never intersect. The notation that is used to describe two parallel lines l 1 and l 2 is l 1 l2. l 1 l 2 6. Plane: A plane is a flat surface with no thickness and no boundaries. Think: a plane looks like a sheet of paper that extends infinitely in all directions. 7. Angle: An angle (usually given in degrees ) is formed by two rays with the same endpoint called the vertex of the angle. The rays are called the sides of the angle. The notation that is used to describe the angle formed below is CAB (angle C-A-B), or BAC (angle B-A-C), or simply A (angle A). Note: m A means the measure of angle A. C A B MAT 107 Career Math Angles & Triangles 150

3 8. Right Angle: A right angle measures 90. The box in the corner indicates a right angle. 9. Straight Angle: A straight angle measures C A B 10. Complementary Angles: Complementary angles are two positive angles whose sum is 90. In the figure below, CAB and DAC are complementary. D C A B 11. Supplementary Angles: Supplementary angles are two positive angles whose sum is 180. In the figure below, CAB and DAC are supplementary. C D A B 12. Acute Angle: An acute angle is an angle whose measure is between 0 and 90 (exclusive). In the figure below, CAB is an example of an acute angle. C A B MAT 107 Career Math Angles & Triangles 151

4 13. Obtuse Angle: An obtuse angle is an angle whose measure is between 90 and 180 (exclusive). In the figure below, CAB is an example of an obtuse angle. C A B 14. Adjacent Angles: Two angles that have a common vertex and a common side but have no interior points in common are adjacent angles. In the figure below, CAB and DAC are adjacent angles. D C A B 15. Vertical Angles: Two angles that are on opposite sides of the intersection of two lines are vertical angles. Vertical angles are equal in measure. In the figure below, A and B are vertical angles. Note: C and D are also vertical angles. A C D B l 1 l Alternate Interior Angles: Alternate interior angles are equal in measure. In the figure below, l 1 l2 and t is a transversal. A and B are alternate interior angles. Note: C and D are also alternate interior angles. E G l 1 A C D B H F l 2 t MAT 107 Career Math Angles & Triangles 152

5 17. Alternate Exterior Angles: Alternate exterior angles are equal in measure. In the figure on the previous page, l 1 l2 and t is a transversal. E and F are alternate exterior angles. Note: G and H are also alternate exterior angles. 18. Corresponding Angles: Corresponding angles are equal in measure. In the figure below, l and t is a transversal. A and H are corresponding angles. Note, there are three 1 l 2 more sets of corresponding angles: E and D, C and F, and G and B. E G l 1 A C D B H F l 2 Example 1: For the following, use your knowledge of angles to find the measure of angle x: t a) Given, L x O m LON = 90 b) M x + 18 N 7x 3 4x + 18 c) If l 1 l2, find m a and m b : 47 l 1 a b l 2 MAT 107 Career Math Angles & Triangles 153

6 Solution to Example 1: a) Since LON = 90 we can say that LOM and MON are complementary angles, that is, LOM + MON = 90. So we get the equation (x) + (x + 18) = 90. Solving for x we get x = 36. Answer: 36 b) Two intersecting lines form vertical angles, which are equal in measure. So we get the equation (7x 3) = (4x + 18). Solving for x we get x = 7. Answer: x = 7 c) l 1 l2 is cut with a transversal, therefore we have alternate interior angles. m b = 47 by alternate interior angles. Now, b and a are supplementary angles, that is, their sum is 180. m a = = 133. Answer: m a = 133 and m b = 47 Constructing an angle: An angle is determined by rotating a ray about its endpoint (vertex). The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side. An angle in standard position on a Cartesian plane has the origin as its vertex, and the initial side coinciding with the positive x-axis. Positive angles: A positive angle is generated by a counterclockwise rotation. See the figure below. y-axis terminal side origin Initial side x-axis Negative angles: A negative angle is generated by a clockwise rotation. See the figure below. y-axis origin Initial side x-axis terminal side MAT 107 Career Math Angles & Triangles 154

7 Note: In applications, we will not need to use negative angles; however, if we wish to define in trigonometry the six trigonometric functions and their domain, it is important to have an understanding of negative angles. Also, if we are constructing angles larger than 360, we complete the circle and then start over. For instance the angle 475 can be constructed by completing one full counterclockwise circle then an additional 115. Example 2: a) Construct the angle 135 in standard position. b) Construct the angle -390 in standard position. Solution to Example 2: a) Since 135 is positive, we will rotate the terminal side counterclockwise. 135 = y-axis terminal side 135 origin Initial side x-axis b) Since -390 is negative, we will rotate the terminal side clockwise. Also, we are making one full circle, 360, plus an addition 30. y-axis -390 origin Initial side x-axis terminal side MAT 107 Career Math Angles & Triangles 155

8 When we construct angles, the angle either lies in a quadrant or on an axis. There are four quadrants (I, II, III, IV) starting with I in the upper right corner of the Cartesian plane and working counterclockwise. y-axis Quadrant II Quadrant I Quadrant III Quadrant IV x-axis We can determine in which quadrant an angle lies by locating the angles terminal side. An angle lies in the quadrant in which its terminal side rests. If an angle lies on an axis, we say that the angle is quadrantal. 0, 90, 180, 270, 360, etc. are examples of quadrantal angles. II. D M S, Radians, and Arc Length So far up to this point we have considered angles that are integers given in degree measure. In this section we will address the notion of fractional degrees as well as a different unit of measure for an angle called a radian. Angles that include measurements that are a fraction of a degree can be written in two ways. (1) In decimal form, such as , or (2) In degrees, minutes, seconds form, such as Degrees, minutes, seconds (D M S ) is a base 60 system (like how time works). 1 degree is equal to 60 minutes, and 1 minute is equal to 60 seconds. To convert back and forth between decimal degrees and D M S we need to use dimensional analysis, and the conversion factors below. Many calculators have the capability of converting to and from D M S. Consult your calculators user manual to help determine if your calculator is capable of doing this. Degrees, Minutes, Seconds (D M S ): 1 counterclockwise revolution = = 60 (minutes) 1 = 60 (seconds) 1 = 3600 Example 3: Convert to decimal degree form. MAT 107 Career Math Angles & Triangles 156

9 Solution to Example 3: To convert to decimal degrees we must use the conversion factors for minutes to 6' 1 21" 1 degrees and seconds to degrees. 50 6' 21" = ' + 21" = ' " Notice that the minutes and seconds reduce to give us Answer: Example 4: Convert to D M S form. Solution to Example 4: To convert to D M S form we must use the conversion factor for degrees to minutes ' first: = = 21 + = '. Now since we have a 1 1 fraction of a minute, we will convert minutes to seconds: 0.36' 60" ' = ' + = 21 15'21.6". Answer: ' Many formulas that involve angle measure, including the arc length formula, require that the angle be given in radians. Next we will learn how to convert between degrees and radians. Radian: One radian is the measure of a central angle θ that intercepts an arc S equal in length to the radius r of the circle. See the figure below. y-axis θ r S = r x-axis To calculate how many radians in one counterclockwise revolution we need to know two formulas: (1) The formula for finding the circumference of a circle (C = 2πr), and (2) The formula for finding the arc length of a circle. Arc length: For a circle of radius r and a central angle of θ (given in radians), the arc length, S, can be found by using the formula: S = rθ MAT 107 Career Math Angles & Triangles 157

10 The arc length formula says that the central angle θ (in radians) is the ratio r S (or arc length divided by the radius). Now we can develop the relationship between degrees and radians as follows: Given the arc length formula S = rθ, and the formula for circumference of a circle C = 2πr, we can say that the arc length of a full circle (central angle = 360 ) is the same as the circumference of the circle. So we can equate the two formulas giving us: rθ = 2πr. Because we want to know how many radians are in one full counterclockwise rotation (360 ), we will solve for θ. Dividing the equation rθ = 2πr by r we get θ = 2π radians. So there are 2π radians in 360. Relationship between Degrees and Radians: 360 = 2π radians 180 = π radians π 1 = radians radian = 180 degrees π Example 5: Convert 60 to radians. Leave your answer in reduced fraction form, in terms of π. Then approximate your answer to the nearest hundredth. Solution to Example 5: Using dimensional analysis, and the conversion factor 180 = π radians, we can convert as follows: 60 π radians π π = radians. Answer: radians radians Example 6: Convert 6 π radians to degrees. Solution to Example 6: Since 180 = π radians, we can simply substitute 180 in for π in the angle 6 π. Once we reduce the fraction we get 30. Answer: 30 MAT 107 Career Math Angles & Triangles 158

11 Example 7: A circle has a radius of 4 in. Find the length of the arc intercepted by a central angle of 240. Approximate your answer to the nearest hundredth. Solution to Example 7: We will use the formula for arc length to solve this problem, but first we must convert 240 to 4π radians. In doing so we get 240 = radians. Now, we can substitute r = 4, and 3 4π 4π θ = radians in the formula S = rθ to get: S = inches. 3 3 Answer: The arc is inches III. Bearings In applications involving surveying and navigation a bearing is usually needed. A bearing describes a line as heading north or south, and deflected some number of degrees toward the east or west. A bearing, therefore, will always have an angle less than 90. For example, the bearing S 40 W (read: south 40 degrees west ) tells us to start on the fixed south line, then rotate that line 40 toward the west direction. The figure below illustrates the bearing S 40 W. North West East 40 South Example 8: Megan starts at a point and goes in the direction shown in the figure below. What is her bearing? MAT 107 Career Math Angles & Triangles 159

12 Solution to Example 8: Megan is in the northeast quadrant (quadrant I). Beginning at the fixed north line, count the degrees from north in the east direction, for a total of 70. Megan's bearing is written as N 70 E (North 70 East). Answer: N 70 E Note that a straight line has two bearings. For example, the line in Example 8 can be extended down into the SW quadrant (quadrant III). If Megan wished to turn around and walk back in the direction she came from, her bearing would be S 70 W. Another way to describe direction is with the use of an azimuth. An azimuth is the direction measured in degrees clockwise from north on an azimuth circle. An azimuth circle consists of 360 degrees. Ninety degrees corresponds to east, 180 degrees is south, 270 degrees is west, and 360 degrees and 0 degrees mark north. See Figure 6.1 below. Figure 6.1: An azimuth on a plot of land The word "bearing" is sometimes used interchangeably with azimuth to mean the direction (the degree reading) from one object to another. This usage is correct only in the first (NE) quadrant between 0 and 90. Azimuth is often used to denote wind direction. Traditionally, wind direction is reported as one of eight compass points (N, NE, E, SE, S, SW, W, NW). However, plots of wind and speed direction often give an azimuth value using 0 and 360 for north, as described above. Winds are named according to the direction from which they are blowing. For instance, a westerly MAT 107 Career Math Angles & Triangles 160

13 wind blows from the west (or 270 ), while a southeasterly wind blows from the southeast (135 ). Example 9: What is the azimuth reading for a bearing of S 23 W? Solution to Example 9: S 23 W can be described with the following drawing: North West East 23 South The azimuth for the bearing S 23 W starts from north then rotates clockwise until it reaches the directional line. North West 203 East South Since the angle of clockwise rotation to reach the desired direction is = 203, the azimuth is 203. Answer: 203 MAT 107 Career Math Angles & Triangles 161

14 IV. Properties of Triangles In this section we will begin by defining a triangle, then, we will investigate different types of triangles that are defined by the measures of their interior angles as well as sides. Three lines intersecting in three distinct points (vertices) forms a triangle. The following three properties hold true for all triangles. 1. The angles within the region enclosed by the triangle are interior angles. Their sum is 180. In the figure below, m A + m B + m C = 180. C A B 2. The sum of the lengths of any two sides of a triangle must exceed the length of the remaining side. This is known as the triangle inequality. 3. Larger angles are opposite larger sides. Example 10: Given that m and m y = 110, find a m b : y a b Solution to Example 10: Since angle y is 110 we can find one of the interior angles of the triangle that is formed by supplementary angles. Now we have two interior angles of the triangle, 70 and 90 (since there is a right angle). Now we can find b. Since the sum of all of the interior angles of a triangle is 180, we can subtract the sum of 70 and 90 from 180 to get b. b = 20. Now to find a, we know that angles a and b are supplementary, so a = 160. Answer: a = 160, b = 20 MAT 107 Career Math Angles & Triangles 162

15 Types of Triangles: Isosceles Triangle: Exactly two sides have equal length and the angles opposite the equal sides are of equal measure. Equilateral Triangle: All sides are of equal length and all angles of equal measure (60 ). Scalene Triangle: A scalene triangle is a triangle in which no two sides are of equal length and no two angles are of equal measure. In other words, no angles or sides are the same in the scalene triangle. Acute Triangle: A triangle in which all three interior angles are acute angles. Obtuse Triangle: A triangle that contains one obtuse interior angle. Right Triangle: A triangle that contains a right interior angle. Isosceles Right Triangle: An isosceles triangle that contains a right interior angle. V. Properties of Right Triangles In trigonometry (meaning triangle measure ), much emphasis is placed on solving right triangles (finding the measure of all angles and measure of all sides). In that process, the Pythagorean Theorem is used extensively as well as properties of triangles and (isosceles right) triangles. Since many real world applications involve triangulation, in this section we will learn the fundamentals of finding lengths of sides of right triangles. Pythagorean Theorem: In any right triangle with hypotenuse c, and legs a and b then 2 2 a + b = c 2 Note: Any values a, b, and c satisfying the Pythagorean Theorem are sides of a right triangle and are called Pythagorean triples. MAT 107 Career Math Angles & Triangles 163

16 Example 11: In the right triangle given below, find the length of side x x Solution to Example 11: Because we have a right triangle we can use the Pythagorean Theorem. Substituting a = 7, b = x, and c = 25 into the Pythagorean Theorem, we get: ( 7 ) 2 + ( x ) 2 = ( 25 ) 2 or 49 + x 2 = 625. Solving for x, we get: x 2 = 576. Taking the square root of both sides gives us x = 24. Answer: x = 24 Special Right Triangles: There are two special right triangles that we will consider in this section: (a) Triangle, and (b) Triangle (Isosceles Right Triangle). a) Triangles: In a triangle the ratio of sides is given to be x : x 3 : 2x. This means that if the side opposite the 30 angle is 5 inches, then the side opposite the 60 angle is 5 3 and the hypotenuse is 10. See Figure 6.2 below. 2x 60 x 30 x 3 Figure 6.2: triangle Example 12: If, in a triangle, the length of the side opposite the 30 angle is 4 inches, find the lengths of the other two sides. Solution to Example 12: The ratio of sides (smallest to largest) in a triangle is x : x 3 : 2x. Using this fact, and given the length opposite the 30 angle is 4 in., we know the ratio of sides then is 4 : 4 3 : 8. So the lengths of the other two sides are 4 3 inches and 8 inches. Answer: 4 3 inches and 8 inches MAT 107 Career Math Angles & Triangles 164

17 b) Triangles (Isosceles Right): In a triangle the ratio of sides is given to be x : x : x 2. This means that if the side opposite a 45 angle is 5 inches, then the remaining two sides are 5 inches and 5 2 inches. x 2 45 x 45 x Figure 6.3: triangle Example 13: The base path of a baseball field is a square with side 90 ft. What is the distance from home plate to second base? Round your answer to the nearest tenth of a foot. Solution to Example 13: Since the base path of a baseball field is a square with side 90 ft, the distance from home plate to second base cuts the square in half, forming a isosceles right triangle. So the ratio of sides is x : x : x 2. Since x = 90 ft, the ratio of sides is 90 : 90 : So the distance from home plate to second base is ft. Answer: ft VI. Similar Triangles Similar Triangles: If ABC ~ DEF then: 1. The ratio of lengths of corresponding sides are equal. We say that they are proportional. This means that if we wish to find the length of a missing side given two similar triangles, we can set up a proportion and solve. 2. The measures of corresponding angles are equal (or congruent). 3. The ratio of corresponding heights is equal to the ratio of corresponding sides. Also, if two angles of one triangle are congruent to two angles of another triangle, then we can say that the two triangles are similar. Note: The above holds true for similar polygons. We will discuss polygons in Module VII. MAT 107 Career Math Angles & Triangles 165

18 Congruent Triangles: ( means congruent ) ABC DEF if: 1. The three sides of one triangle are equal in measure to the corresponding three sides of a second triangle. 2. Two sides and the included angle of one triangle are equal in measure to two sides and the included angle of a second triangle. 3. Two angles and the included side of one triangle are equal in measure to two angles and the included side of a second triangle. Example 14: Let ABC ~ DEF and the height of DEF is 1.1. a) Find DE b) Find the height of ABC C F A 2.7 B D E Solution to Example 14: a) Since triangle ABC is similar to triangle DEF, we know that the lengths of the corresponding DE 2.4 sides are proportional. Setting up a proportion, to solve for DE we get =. Solving for DE we get DE = Answer: DE = 0. 9 b) Since triangle ABC is similar to triangle DEF, we know that the ratio of corresponding heights is equal to the ratio of corresponding sides. Setting up a proportion by using the height of triangle DEF given to be 1.1, we can solve for the height of triangle ABC, we ll call it h. The h 7.2 proportion becomes =. Solving for h we get h = 3.3 Answer: h = Example 15: A tree casts a shadow 22 ft long while at the same time a 5 ft 10 in. man casts a shadow 7 ft 2 in. long. Estimate the height of the tree to the nearest inch. MAT 107 Career Math Angles & Triangles 166

19 Solution to Example 15: Since the triangle formed from the tree and its shadow is similar to the triangle formed from the man and his shadow we can set up a proportion, but first let s write all of the measurements in inches. 22 ft = 264 in., 5 ft 10 in. = 70 in., and 7 ft 2 in. = 86 in. Using these h 70 figures with h = height of the tree, our proportion is: 264 =. Solving for h we get, h is 86 approximately 215 inches or 17 ft 11 in. Answer: 17 ft 11 in. VII. Finding Lengths Given Angles In this section we will use a trigonometric function, specifically the tangent function, to find lengths of missing sides of a right triangle. To keep it simple, we will not define any of the other five trigonometric functions. Tangent Function: In any right triangle given an interior angle θ and the sides opposite and adjacent to θ, we can define the following equation: tan θ = side opposite angle θ side adjacent angle θ tan means tangent and it can be found on most scientific calculators. Figure 6.4 below illustrates the tangent function. y θ x Figure 6.4: tan θ = y x Example 16: From a point on level ground 50 feet from the base of a building, the angle of elevation from your foot to the top of the building is 74. Find the height of the building. MAT 107 Career Math Angles & Triangles 167

20 Solution to Example 16: The situation described gives the following triangle: building θ = ft y h Substituting the values given into the equation tan θ = we get: tan 74 =. Solving for h we get x 50 h = 50 tan So the height of the building is approximately ft. Answer: ft Note: In the previous example if we were asked to find the distance from our feet to the top of the building, after finding the height of the building, we could use the Pythagorean Theorem to find the length of the hypotenuse of the right triangle that is formed. VIII. Finding Angles Given Slope Slope refers to the upward or downward angle, grade, or pitch, of an incline. Slope is defined to be the change in vertical distance divided by the change in horizontal distance, otherwise known as rise over run. If we re driving on I-70 in Colorado heading westbound just out of 6 the Eisenhower Tunnel we will approach a sign indicating a 6% grade. Since 6% means, 100 this sign indicates that we will be descending a vertical distance of 6 feet for every horizontal distance of 100 feet. See Figure 6.5 below. 6 ft car 100 ft Figure 6.5: 6% grade Slope can also be expressed as an angle. The angle of elevation is the angle formed from the horizontal to a path or line of sight. See Figure 6.6 on the next page. MAT 107 Career Math Angles & Triangles 168

21 Line of sight θ = angle of elevation Horizontal Figure 6.6: Angle of Elevation y Recall the tangent function in the previous section is given by the equation: tan θ =. x Because slope is the vertical change divided by the horizontal change, tan θ is equal to the slope of the incline, and the angle θ describes the incline, or slope, in degrees. Conversions rise between slope percent, 100, and slope angle can be done using a scientific calculator run and the inverse tangent function ( tan 1 or arctan). Essentially, the slope angle is the inverse tangent of the slope percent (with slope percent expressed in decimal form), that is, the angle θ 1 y can be found using the equation θ = tan. See Figure 6.7 below. x y θ x Figure 6.7: θ = tan 1 y x The figure on the next page (Figure 6.8), illustrates the relationship between slope percentage and angle measure in degrees. MAT 107 Career Math Angles & Triangles 169

22 Figure 6.8: Slope in terms of angles and percents Example 17: An airplane taking off on a runway climbs at a slope of 32.5%. What is the airplanes angle of elevation? Solution to Example 17: Using the equation = tan y x 1 y θ with = 0.325, we get ( ) x θ = tan Answer: 18 Basic Examples (to be combined with emphasis area): In problems 1 8, name each angle s complement and supplement, if possible MAT 107 Career Math Angles & Triangles 170

23 In problems 9 17, use the given information and the figures to answer the question. 9. Given that 1 2, a b 1 c d e f g h 2 a) Name any pair of alternate interior angles in the above figure. b) Find the sum of the measures of angles b and h. c) Name any pair of vertical angles in the above figure. d) Find the difference of the measures of angles a and e. 10. Find the measure of angle y. t 109 y 11. Find the measure of angle x. 2x 111 x Use the diagram to the right to solve for the missing angle measure. a) If m 1 = 48 find the measure of 2 b) If m 2 = 123 find the measure of 4 MAT 107 Career Math Angles & Triangles 171

24 13. Use the diagram below to solve for x and y. 14. Given that m b = 122 and m x = 31, find the measures of angles a and y. a b x y 15. In the figure below, the measure of angle a is 9 less than half the measure of angle b. Find the measure of angle a and angle b. a b 16. In the figure below, find the measure of angle A A 17. In the figure below, find the measure of angle x. x 67 MAT 107 Career Math Angles & Triangles 172

25 In problems 18 24, 1 2 in the figure below. Use the figure to find the missing angles. a b 1 c d e f 128 g Find m a. 19. Find m b. 20. Find m c. 21. Find m d. 22. Find m e. 23. Find m f. 24. Find m g. t In problems 25-28, use properties of triangles to answer the questions. 25. An angle of an isosceles triangle is given to be 36. Find the measures of the remaining two angles. 26. Find any angle of an equilateral triangle? 27. Is it possible for an isosceles right triangle to have an interior angle of 95? Explain. 28. If in a scalene triangle you are only given one angle measure of 44, is it possible to find either of the two remaining angles? Explain In problems 29 32, sketch each angle in standard position In problems 33 38, determine which quadrant the given angle lies π 35. radians π radians π radians MAT 107 Career Math Angles & Triangles 173

26 In problems 39 42, convert from D M S (Degrees, Minutes, Seconds) to decimal degrees Add and write your answer in decimal degree form: In problems 44 47, convert from decimal degrees to D M S (Degrees, Minutes, Seconds). Round each answer to the nearest second In problems 48 51, convert each angle given in degrees to radians. Leave your answer as a reduced fraction in terms of π In problems 52 55, convert each angle given in degrees to radians. Round your answer to the nearest thousandth In problems 56 59, convert each angle given in radians to degrees π radians 4 3π radians 5 7π radians 6 8π radians 9 MAT 107 Career Math Angles & Triangles 174

27 In problems 60 63, convert each angle given in radians to degrees. Round each answer to the nearest thousandth radians radians radians radians In problems 64 65, answer the following questions. 3π 64. Find the supplement of an angle with a measure of radians. 5 10π 65. Circle the best answer below. radians is this type of angle: 9 Right Obtuse Acute Straight None of these In problems 66 73, use the arc length formula to answer the questions. Round each answer to the nearest hundredth. 66. Find the arc length on a circle with radius 280 m intercepted by a central angle of 3.5 radians. 67. Find the arc length on a circle with radius 129 cm intercepted by a central angle of 2.3 radians. 68. The arc length of a circle with radius 2676 miles is 2007 miles. What is the central angle, in degrees? 69. The arc length of a circle with radius 12,320 km is km. What is the central angle, in degrees? 70. Find the radius of a circle with an arc length of 22 in. intercepted by a central angle of Find the arc length on a circle with a radius of 15 in. intercepted by a central angle of Assuming that the Earth is a sphere of radius 6378 kilometers, what is the difference in latitudes of Syracuse, New York and Annapolis, Maryland, where Syracuse is 450 kilometers due north of Annapolis? Write your answer in degrees. 73. The Earth has a radius of 6378 km. A Geostationary satellite system orbits the Earth at a distance 40,000 km above Earth. If the satellite orbits the Earth in a perfect circle, what distance has the satellite travelled after one full rotation around the Earth? In problems 74 76, write each given bearing as an azimuth. 74. N 56 W 75. S 22 E 76. N 79 E MAT 107 Career Math Angles & Triangles 175

28 In Problems 77 79, write each given azimuth as a bearing In problems 80 84, identify the type of triangle formed by the following sets of lengths of sides as equilateral, isosceles, isosceles right, right, , or no triangle , 8, , 3, , 8 3, , 7, , 6, 10 In problems 85 92, use the Pythagorean Theorem to answer the following questions. Round any decimal answers to the nearest hundredth. 85. Find the length of the hypotenuse of a right triangle with legs measuring 5 ft and 12 ft. 86. Find the length of the missing leg of a right triangle with hypotenuse 10 m and another given leg of 6 m. 87. Find the length of the altitude of a triangle with all sides of length 10 in. 88. If the top edge of a 15 foot ladder reaches a window sill 12 feet above level ground, how far from the house is the ladder placed? 89. For maximum safety the distance between the base of a ladder and building should be onethird of the length of the ladder. If a window is 20 ft above level ground, how long a ladder is needed to meet the safety condition? 90. The hypotenuse of a right triangle is 7 m long. One leg is 3 m longer than the other. Find the lengths of the legs. Round your answer to the nearest tenth. 91. The length of a rectangle is 3 m greater than the width. The area is 108 square meters. Find the length and width. 92. A rectangular plot measures 16 meters by 34 meters. Find, to the nearest meter, the distance from one corner of the plot to the corner diagonally opposite. 93. The glass for a window is 2.3 meters wide. The doorway is 0.9 meters wide. About how high must the doorway be in order for the contractor s helpers to get the glass through the doorway? In problems 94 98, use your knowledge of special right triangles to answer the questions. Round any decimal answers to the nearest hundredth. 94. Find the length of a side of an equilateral triangle given its altitude is six inches long. MAT 107 Career Math Angles & Triangles 176

29 95. Find the total length of the two wire braces for the two poles given in the figure below. 4 m m 96. A 24 ft. ladder leans against a building and makes an angle of 60 with level ground. How far up on the building does the top of the ladder reach? 97. Find the length of the hypotenuse of an isosceles right triangle if each leg is 32 inches long. 98. If, in a triangle, the length of the hypotenuse is 14 centimeters, find the length of the side opposite the 60 angle. In problems , use your knowledge of similar triangles and the diagram to answer the questions. 99. In the figure below BD AE. If BD = 6 in., AC = 15 in. and AE = 8 in. Find BC. C B D A E 100. In the following figure: MQ and NP intersect at O. NO = 16 in., MN = 12 in., PQ = 4 in. and 2 MQ = 26 in. Find the perimeter of OPQ 3 M N O P Q MAT 107 Career Math Angles & Triangles 177

30 101. A ten-foot lamppost casts a shadow sixteen feet long and at the same time a person casts a shadow six feet long. How tall is the person? 102. With 100 feet of string out, a kite is 64 feet above ground level. When the girl flying the kite pulls in 40 feet of string, the angle formed by the string and the ground does not change. What is the height of the kite above the ground after the 40 feet of string have been taken in? y In problems , use the equation tan θ = to answer the questions. Round any x decimal answers to the nearest hundredth A person stands 34 ft. from the base of a building on level ground. The angle of elevation from the person s foot to the top of the building is 82. Find the height of the building A person stands some distance from the base of a 45 ft pole on level ground. If the angle of elevation from the person s foot to the top of the pole is 34, how far is the person from the pole? 1 y In problems , use the equation θ = tan to answer the question. Round any x decimal answers to the nearest hundredth The slope of mountain is. What is the angle of elevation? The slope of a staircase is. What is the angle of elevation? The grade of a roadway is 8%. At what angle is the roadway descending? 108. The pitch of a roof is 58%. What is the roofs angle of elevation? 1. APPLICATIONS FOR FIRE SCIENCE FUN FACT: Firefighters use angles every day. Angles can help determine position using a compass or on a map. The slope of an incline can be expressed as an angle, and can be a deciding factor in whether a dozer can drive safely up the incline. Angles also provide information about the steepness of slope to determine potential effects on fire spread. In problems , use the given figure below to answer the questions. MAT 107 Career Math Angles & Triangles 178

31 109. Jake is walking on the line in the sketch going southerly. What is his bearing? 110. If Jake turns around in the example above, what will his bearing be? In problems , answer the questions. Round each answer to the nearest hundredth Given that a slope percent is 60%. What is the slope angle? 112. Given that a slope percent is 250%. What is the slope angle? 113. A fire is spreading up a hill that has a slope of 7 2. What is the angle of elevation? 114. A fire is spreading up a hill that has a slope of What is the angle of elevation? 115. Use the figure below to estimate the slope angle A hill has a slope of The height of the hill is 15 feet. What is the horizontal distance? 117. Tom has a clinometer (a device used to measure slope) on a 5.0-foot pole. He sights another 5.0-foot pole that is 24.0 feet away (horizontal distance). He gets a reading of 20.0 slope percent (see figure below). a) What is the slope? b) What is the slope angle (angle of elevation)? c) What is the vertical distance? 118. Find the slope distance for the vertical and horizontal distances illustrated in the figure below. MAT 107 Career Math Angles & Triangles 179

32 2. APPLICATIONS FOR GRAPHIC DESIGN/PROFESSIONAL PHOTOGRAPHY FUN FACT: In most photography, longer focal length or lower optical power is associated with larger magnification of distant objects, and a narrower angle of view. Conversely, shorter focal length or higher optical power is associated with a wider angle of view. We can use the Pythagorean Theorem to find the normal focal length of a camera, where a and b are the sensor dimensions, and c is the normal focal length. In problems , use the information given in the table and the Pythagorean Theorem to fill in the missing sensor dimension or normal focal length in each column. Camera Sensor Sensor Normal Dimension Dimension Focal Length 119. Canon A100, A mm 3.4 mm 120. Nikon Coolpix 100, 300 Casio QV-8000SX 4.8 mm 3.6 mm 121. Nikon Coolpix 2500, 3500 Sony DSC-P31 Pentax Optio 230, 330GS Canon A40 Olympus C-730 Minolta Dimage X, Xi 4.0 mm 6.6 mm 122. Nikon Coolpix mm 8.0 mm 123. Nikon Coolpix 995, 4300, 4500, 5400 Canon Powershot G2, G3, G5, S30, S40, S45, SD900 Kodak DX3900, 4900 Pentax Optio 330RS, 430RS Olympus C-5050, C-5060, C-8080 Ricoh GR-D, GX Nikon Coolpix 5000, 5700 Sony DSC-F717 Minolta 7i, 7Hi 125. Olympus E-1 with the 5 megapixel chip Olympus E-300 with an 8 megapixel chip also made by Kodak. Now also E-500, E-330 and E mm 8.9 mm 8.8 mm 11.0 mm 18.0 mm 13.5 mm 126. Canon D30, D60, 10D 22.7 mm 15.1 mm 127. Nikon D1, D1H, D1X, D mm 28.4 mm 128. Pentax *ist D 15.7 mm 28.3 mm 129. Canon 1D 28.7 mm 33.8 mm 130. Advanced photo system film cameras (various crops in camera) APS-H here at 16:9 ratio mm film cameras. Full frame digital SLRs such as Contax N, Canon 1Ds, Kodak DCS14n 30.2 mm 34.5 mm 36.0 mm 24.0 mm 132. Medium format 120 roll film 56.0 mm 41.5 mm MAT 107 Career Math Angles & Triangles 180

33 133. What lens is the closest to the mathematical normal field of view for a 4x5 inch view camera? a. 90 mm b. 165 mm c. 210 mm d. 300 mm 3. APPLICATIONS FOR INTEGRATED ENERGY TECHNOLOGY In problems , answer the question. Round your answer to the nearest hundredth In March at 9 am in Gypsum, Colorado, the angle of elevation from the horizontal to the sun is 30. How long of a shadow would a 20 foot tree make at this time? 135. In February at 12 pm in Rifle, Colorado, the angle of elevation from the horizontal to the sun is 38. How long of a shadow would a 32 foot tree make at this time? 136. In June at 3 pm in Glenwood Springs, Colorado, the angle of elevation from the horizontal to the sun is 50. How long of a shadow would a 50 foot building make? 137. In April at noon in Eagle, Colorado, the angle of elevation from the horizontal to the sun is 62. How long of a shadow would a 72 foot building make? 138. In December at noon in Gypsum, Colorado, a 48 foot tree makes a 100 foot shadow. What is the angle of elevation to the sun? 139. In June at 1 pm in Eagle, Colorado, a 27 foot tree makes a 10 foot shadow. What is the angle of elevation to the sun? 4. APPLICATIONS FOR PROCESS TECHNOLOGY Recall from Module I, we first defined the hydraulic gradient in terms of the head loss of two points and the distance between them. In Module III, we began the process of finding the hydraulic gradient given three wells by partitioning the line between the highest and lowest head into 0.1 increments. We will now finish the process and completely find the hydraulic gradient given only the heads of three wells. Below is an example of the process that you will be using in the following exercises to find the flow line and the hydraulic gradient. Practice Example: Well A = 10.4 meters head, Well B = 10.0 meters head, and Well C = 9.9 meters head in the given figure below. The distance between well A and well B is 500 meters and the distance between well A and well C is 500 meters (forming a right angle). Determine the direction of the flow of groundwater; then calculate the hydraulic gradient. A 500 m B 500 m C MAT 107 Career Math Angles & Triangles 181

34 First, since well A has the highest head (10.4 m) and well C has the lowest head (9.9 m) we will only consider these two wells to partition in increments of 0.1 m. Then, since there are five increments of 0.1, we take 500 meters and divide by 5, giving us 100 m per 0.1 m increment. A = m m m m m C = 9.9 B = 10.0 Second, we will draw a line from the third well (the well with the middle head) to the point on the line between the first two wells where the head is the same as that in the third well, we ll call this point D. This line is called an equipotential line. This means that the head anywhere along the equipotential line should be constant (in this case 10.0). Groundwater will flow in the direction perpendicular to this line. A = B = equipotential line 10.1 D = 10.0 C = 9.9 Third, draw a line perpendicular to the equipotential line through the well with the highest head. This line is called the flow line. The ground water flow is in a direction parallel to this line. A = flow line B = equipotential line 10.1 D = 10.0 C = 9.9 MAT 107 Career Math Angles & Triangles 182

35 With triangle ABD formed we are ready to (a) calculate angle α, and (b) calculate the distance r given in the figure below. A 500 r = flow line 400 equipotential line α D B Since we have a right triangle with the measurement of the legs given, we can use the tangent function to calculate angle α. It is important to note that we would not be able to calculate the angle α in this manner without the initial wells forming a right angle. If they did not form a right angle we would need to use formulas in Trigonometry for solving oblique triangles called the Law of Sines or Law of Cosines, which we will not discuss in this class. Now, by using right triangle ABD we know that tan α = y x, or in this case tan α =, therefore tan 1 α = We now need to define another trigonometric function that will allow us to calculate r. In any right triangle with sides x, y, and r, and central angle θ, below. y sin θ =. See the figure 6.9 r r y θ Figure 6.9: x sin θ = To calculate the distance r, we will consider the right triangle formed with well A, the point D, and the flow line. y r MAT 107 Career Math Angles & Triangles 183

36 A 400 m r = flow line α = D r Using the sine function we get: sin =. Solving for r we get: 400 r = 400sin m h2 h1 Finally, to calculate the hydraulic gradient we use the formula: hydraulic gradient = r with h 2 = highest head and h 1 = head of the equipotential line, and r = flow line Hydraulic gradient = = In problems , answer the following questions. Round r to the nearest hundredth, and round the hydraulic gradient to two significant figures Well A = 15.4 meters head, Well B = 9.0 meters head, and Well C = 9.8 meters head in the given figure below. The distance between well A and well B is 300 meters and the distance between well A and well C is 250 meters. Determine the distance r of the flow line, then calculate the hydraulic gradient. 300 m A B 250 m C 141. Well A = 13.7 meters head, Well B = 12.2 meters head, and Well C = 10.8 meters head in the given figure below. The distance between well A and well B is 570 meters and the distance between well A and well C is 550 meters. Determine the distance r of the flow line, then calculate the hydraulic gradient. 570 m A B 550 m C MAT 107 Career Math Angles & Triangles 184

37 142. Well A = 8.9 meters head, Well B = 8.0 meters head, and Well C = 8.5 meters head in the given figure below. The distance between well A and well B is 275 meters and the distance between well A and well C is 325 meters. Determine the distance r of the flow line, then calculate the hydraulic gradient. 275 m A B 325 m C FUN FACT: When constructing tanks and other equipment used in the oil industry (as seen below in the photo taken at Colorado Mountain College s West Garfield Campus), oil industry technicians need to be familiar with pipe fitting. The next few exercises focus on terms used in pipe fitting and their relationship to angles, right triangles, and right triangle trigonometry. FUN FACT: A piping offset refers to a section in piping that bends creating an angle other than a straight angle between two (or more) sections of piping. In figures 6.10 and 6.11 below, we see that a piping offset creates a right triangle with the vertical change being defined as the set, the horizontal change being defined as the run, and the hypotenuse being defined as the travel. Figure 6.10: Piping Offset Figure 6.11: Piping Offset Right Triangle MAT 107 Career Math Angles & Triangles 185

38 In problems , use the Pythagorean Theorem, properties of special right triangles, the tangent function, or the inverse tangent function to answer the questions regarding piping offset. Round each answer to the nearest hundredth What is the length of the travel if the run is 24 inches and the set is 18 inches? 144. What is the length of run and travel for a 45 offset with a set of 15 inches? 145. What is the length of set and travel for a 30 offset with a run of 20 inches? 146. What is the length of set for a 36 offset with a 28 inch run? 147. When the set is 18 inches and the run is 24 inches, what is the angle of fitting for a welding offset? 148. When the run is 24 inches and the travel is 30 inches, what is the measure of angle A in the below figure? What is the measure of angle B? 149. What is the length of travel for a 45 offset with a set of 4 feet 8 ¾ inches? 150. Find the length of A and B for the layout shown in the figure below. 45 fittings are to be used The figure below describes a rolling offset. What is the length of the travel and the run if the roll of a 45 offset is 8 inches, and the set is 15 inches? MAT 107 Career Math Angles & Triangles 186

39 FUN FACT: When calculating the length of a simple pipe bend as in figure 6.12 below, we can use the arc length formula: S = rθ, where S is the length of the arc, r is the radius and θ is the central angle in radians. In the figure, D is the angle (90 = 2 π radians), R is the radius, L is the length of the arc, and T is known as the tangent. Note: The tangent, T, carries a different meaning than the trigonometric tangent function that we use to find a missing side or angle. In order for us to find the total length of the pipe (end to end), we must add L + 2T. Figure 6.12: Simple Pipe Bend In problems , use the arc length formula to answer the following questions. Round your answers to the nearest hundredth Find the length of a piece of pipe for a 90 bend with a radius of 40 in. and with two 15-in. tangents Find the length of a piece of pipe for a 90 bend with a radius of 32 in. and with two 8-in. tangents. FUN FACT: A carpenter s square is a tool used to construct or test right angles and is handy in pipe fitting. To create a carpenter s square from wood boards: 1) Nail together two straight pieces of 1 by 2 board to form an L, as shown in figure 6.13 below. 2) Mark off a point 3 feet from the corner on one of the boards. 3) Mark off a point 4 feet from the corner on the other board. 4) Cut another board exactly 5 feet long. Line up this board with the points marked off. MAT 107 Career Math Angles & Triangles 187

40 In problems , answer the following questions Use the Pythagorean Theorem to show that a triangle with lengths 3 feet, 4 feet, and 5 feet form a right triangle Use the Pythagorean Theorem to show that any triangle with lengths that are a constant multiple of a 3, 4, 5 triangle form a right triangle. Hint: Substitute, 3x, 4x, and 5x into the Pythagorean Theorem and then determine if they satisfy. 5. APPLICATIONS FOR SKI AREA OPERATIONS FUN FACT: In constructing terrain parks and half pipes for the air dogs in search of huckfests, engineers need to determine the amount of snow (i.e. volume) that will be needed to create phat jumps. This requires the dimensions of the jumps. Typically, a ski jump or a half pipe is angled off on all sides at a 45 angle (see the figures 6.13 and photos 1 4 below). Using right triangles, one can determine missing lengths that will help in estimating volumes. In this module we will use properties of special right triangles to find missing lengths, and in Module VII, we will find volumes of the same figures. Figure 6.13: Half Pipe Photo 1: Half pipe at Beaver Creek, CO Photo 2: Terrain Park Photo s 3 & 4: Tanner Coulter at the Nature Valley Challenge Denver Big Air Freestyle Ski Event MAT 107 Career Math Angles & Triangles 188

41 In problems , use the Pythagorean Theorem, properties of special right triangles, the tangent function, or the inverse tangent function to answer the questions regarding ski & snowboard freestyle jumps. Round each answer to the nearest hundredth A 22 foot super pipe actually has 25 foot walls by the time you figure in a few feet for the floor of the pipe. If the snow is angled off from the deck to the base of the pipe at a 45 angle in the given figure, calculate the missing lengths x and r: deck 45 r 22 ft x 157. A measurement of 11.5 feet is taken from the base of a half pipe to the edge of the incline. If the angle of elevation from the edge to the deck is 60 in the given figure, calculate the missing lengths y and r: deck r y feet 158. A measurement of 9 feet is taken from the base of a half pipe to the edge of the Incline and the height of the half pipe is measured to be 12 feet. Find the missing length y and the angle of elevation θ: deck y 12 feet θ 9 feet MAT 107 Career Math Angles & Triangles 189