Ready To Go On? Skills Intervention 8-1 Similarity in Right Triangles

Transcription

1 8 Find this vocabular word in Lesson 8-1 and the Multilingual Glossar. Finding Geometric Means The geometric mean of two positive numbers is the positive square root of their. Find the geometric mean of each pair of numbers. If necessar, give the answer in simplest radical form.. and 16 Let be the geometric mean. ()( ) Definition of geometric mean.. 8 and 5 Read To Go On? Skills Intervention 8-1 Similarit in Right Triangles Let be the geometric mean. Find the positive square root. (8)( ) Definition of geometric mean. 00 Find the positive square root. Vocabular geometric mean Finding Side Lengths in Right Triangles Find,, and z. z 8 16 (8)(16) is the geometric mean of 8 and. 18 Find the positive square root. ()( ) 38 is the geometric mean of and Find the positive square root. z ( )(8) z is the geometric mean of and 8. z 8 Find the positive square root. 109 Holt Geometr

2 8 Read To Go On? Problem Solving Intervention 8-1 Similarit in Right Triangles The geometric mean of two positive numbers is the positive square root of their product. To estimate the height of a lighthouse, Henr stands so that his lines of sight to the top and bottom of the lighthouse form a 90 angle. What is the height of the lighthouse to the nearest foot? Understand the Problem 1. How tall is Henr?. What forms the 90 angle? and ft 9 in. 5 ft Make a Plan 3. What do ou need to determine?. Since the length of the altitude to the hpotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hpotenuse, what is the geometric mean of 5 and? 5. How man feet is 9 inches? 6. ft 9 inches ft 7. Let represent the height of the lighthouse above ee level. (length of the altitude ) geometric mean of 5 and. (.75 ) Solve 8. Solve the equation b isolating. 9. The height of the tower to the nearest (.75 ) 5 foot is ft ft Look ack 10. Substitute our solution for into the equation ou wrote in Eercise 7. (.75 ) 5(13) Is our solution approimatel equal to square of the length of the altitude? 110 Holt Geometr

3 8 Read To Go On? Skills Intervention 8- Trigonometric Ratios Find these vocabular words in Lesson 8- and the Multilingual Glossar. Vocabular trigonometric ratio sine cosine tangent Using Trigonometric Ratios to Find Lengths sin opposite leg hpotenuse cos hpotenuse Find each length. Round to the nearest hundredth.. LM _ LM is to the given angle, K. tan adjacent leg L You are given KM, which is to K. 0 M Since the adjacent and opposite legs are involved, use the 35 in. ratio to find LM. K tan tan K opposite leg adjacent leg LM LM Write the trigonometric ratio. Substitute the given values. ( ) tan 0 Multipl each side b.. XZ _ XZ is in. LM You are given YZ, which is the of the triangle. to the given angle, Y. Simplif the epression. Since the opposite side and the hpotenuse are involved, cm use the ratio to find XZ. Y X Z sin sin Y opposite leg hpotenuse 15.3 XZ Write the trigonometric ratio. Substitute the given values. ( ) sin 6 Multipl each side b. cm XZ Simplif the epression. 111 Holt Geometr

4 8 Read To Go On? Skills Intervention 8-3 Solving Right Triangles Solving Right Triangles Given measures can be used to find unknown angle measures or lengths of a triangle, which is known as solving a triangle. To solve a right triangle, ou need to know two side lengths or one side length and a(n) measure. angle Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Method 1 the Pthagorean Theorem, (6.8 ) ( ) Substitute the given values. (3.) m ta n 1 3. Find the square root. 6.8 If tan, then ta n 1 m. m m 90 The acute angles of a right triangle are complementar. Method m ta n If tan, then ta n 1 m. m 90 The acute angles of a right triangle are sin 3. Definition of Sine.. sin 3. Solve for. Substitute for m. sin 11 Holt Geometr

5 8 You can use the inverse tangent ratio to find the measure of unknown angle measures. If tan, then ta n 1 m. carpenter frames a garage which has a roof that peaks at 8 feet from the ceiling joist. If the length of the ceiling joist is 1 feet from the center point to the edge of the garage, what angle is formed b the rafter and the ceiling joist? Round to the nearest degree. Understand the Problem 1. What is the height from the ceiling joist to the peak of the roof?. re the ceiling joists horizontal or vertical? 3. How long is the ceiling joist from the edge to the center of the garage. Make a Plan Read To Go On? Problem Solving Intervention 8-3 Solving Right Triangles. What do ou need to determine? 5. Label the right triangle to show the garage roof. peak rafter height = ft ceiling joist length = ft 6. What angle is formed b the rafter and the ceiling joist? 7. Which trigonometric function involves opposite and adjacent legs of a triangle? Solve 8. If tan, then ta n 1 m. omplete: m ta n 1 1 m 9. What angle is formed b the rafter and the ceiling joist? Look ack 10. Is tan 3 approimatel equal to 8 1? 11. Does our answer seem reasonable? Eplain. 113 Holt Geometr

6 8 Read To Go On? Quiz 8-1 Similarit in Right Triangles Find the geometric mean of each pair of numbers. If necessar, give the answers in simplest radical form and and and 15 8 Find,, and z z z z 1 z z z 7. mountain climbing instructor is setting up a practice mountain and needs to know how much rope he will need. He positions himself so that his line of sight to the top of the cliff and his line of sight to the bottom form a right angle as shown. What is the height h of the practice mountain? 18 ft h 1 5 ft 8- Trigonometric Ratios Use a special right triangle to write each trigonometric ratio as a fraction. 8. sin 5 9. cos tan Holt Geometr

7 8 Read To Go On? Quiz continued Use our calculator to find each trigonometric ratio. Round to the nearest hundredth. 11. sin 3 1. cos tan 5 Find each length. Round to the nearest hundredth PQ 16. JK P 65 Q J 6 K 6 37 L 8 R 8-3 Solving Right Triangles Find the unknown measures. Round lengths to the nearest tenth and angle measures to the nearest degree M L 1 0 N X Z 13 5 Y N Z LN XZ MN XY 0. The wheel chair ramp at a local arena has a ramp length of 0 feet and a rise of 1.7 inches. What angle does the ramp make with the sidewalk? Round to the nearest tenth of degree. 115 Holt Geometr

8 8 Read To Go On? Enrichment Trigonometr and earings In surveing and navigation, directions are usuall given in terms of bearings. bearing measures the acute angle a path or line of sight makes with a fied north-south line. The following eamples eplain this concept. N N N W E W 75 E W 60 E S 37 E means 37 degrees east of south S 37 N 75 W means 75 degrees west of north S N 60 E means 60 degrees east of north S Solve 1. Two lighthouses are 30 miles apart,. hot air balloon is tethered due west lighthouse being due west of of an observation station. The landing pad lighthouse. boat is spotted from is 1 km south of the hot air balloon. the towers, and the bearings from and From the landing pad, the bearing to the are E 1 N and W 3 N, respectivel. observation station is N 63 0 E. How far Find the distance d of the boat from the is the hot air balloon from the observation line segment. station? d miles 1 km 63 0' pad 3. ship leaves port at 8.M. and has a. radio-controlled airplane is 160 meters bearing of S 9 W. If the ship sails at north and 85 meters east of the pilot. If 0 knots, how man nautical miles south the pilot wants the plane to fl directl to and how man nautical miles west will him, what bearings should be taken? the ship have travel b P.M.? 116 Holt Geometr

9 8 Read To Go On? Skills Intervention 8- ngles of Elevation and Depression Find these vocabular words in Lesson 8- and the Multilingual Glossar. Vocabular angle of elevation angle of depression lassifing ngles of Elevation and Depression lassif each angle as an angle of elevation or angle of depression. n angle of elevation is the angle formed b a horizontal line and a line of sight to a point the line. n angle of depression is the angle formed b a horizontal line and a line of sight to a point the line. D is formed b a horizontal line and a line of sight to a point the line. It is an angle of. is formed b a horizontal line and a line of sight to a point the line. It is an angle of. is formed b a horizontal line and a line of sight to a point the line. It is an angle of. D D is formed b a horizontal line and a line of sight to a point the line. It is an angle of. Finding a Measure Using Trigonometric Ratios Find the missing measure. What are the ratios for each trigonometric ratio? sin opposite cos hpotenuse tan opposite 150 What is _ in relation to? What is _ in relation to? Which trigonometric ratio can be used to find? Set up the ratio: tan 150 tan The length of _ is approimatel units long. 117 Holt Geometr

10 8 Read To Go On? Problem Solving Intervention 8- ngles of Elevation and Depression Determine if ou have an angle of elevation of depression before starting to solve a problem. surveor is standing 50 feet from the base of a large tree. The surveor measures the angle of elevation to the top of the tree as 7. How tall is the tree? Round to the nearest foot. Understand the Problem 1. What are ou asked to find?. Do ou have an angle of elevation or depression? 7 50 ft 3. The triangle formed is a right triangle because the forms a 90 angle with the. Make a Plan. You need to determine the side the given angle. 5. In relation to the given angle which side measure are ou given? 6. Which trigonometric ratio can be used to find the height of the tree? Solve 7. Set up the trigonometric ratio: 7 8. Solve the ratio for. 7 tan 7 9. The height of the tree is approimatel ft. Look ack 10. Since ou have the measures of two sides, ou can use the two sides and check to see if the measure of the given angle matches our calculation. tan 50 tan tan 1 Does our result match the given angle measure, 7? 118 Holt Geometr

11 8 Read To Go On? Skills Intervention 8-5 Law of Sines and Law of osines Using the Law of Sines The Law of Sines can be used to solve a triangle if ou are given: two angle measures and an side length (S or S) or two side lengths and a non-included angle measure (SS). Find. Round to the nearest tenth. Looking at the given diagram, are ou given S, S or SS? State the Law of Sines: sin a b Which two proportions should be used? sin a sin sin sin 36 Substitute known value. sin sin Use the ross Products Propert. sin sin Divide to solve for. The length of side is Using the Law of osines The Law of osines can be used to solve a triangle if ou are given: two side lengths and the included angle measure (SS) or 1 three side lengths (SSS). 6 Find. Round to the nearest tenth. Looking at the given diagram, are ou given SS, or SSS? Which formula for the Law of osines should be used? a c c cos What does b equal? What does c equal? Substitute known values into the formula. a 1 ( )( ) cos 6 a a The length of _ is Holt Geometr

12 8 Read To Go On? Skills Intervention 8-6 Vectors Find these vocabular words in Lesson 8-6 and the Multilingual Glossar. Vocabular vector component form magnitude direction equal vectors parallel vectors resultant vector Finding the Magnitude of a Vector Draw vector 3, 3 on a coordinate plane. Find its magnitude to the nearest tenth. Use the origin as the initial point. Then (3, 3) is the point. What is the horizontal change? What is the vertical change? The of a vector is its length. To find the magnitude ou use the Distance Formula. 3, 3 ( 0 ) ( 0 ) 9 Finding the Direction of a Vector The force eerted b a skier on a tow rope is given b the vector, 5. Draw the vector on a coordinate plane. Find the direction of the vector to the nearest degree. Step 1 Draw the vector on the coordinate plane. Use the origin as the initial point. Then (, 5) is the point. What is the horizontal change? What is the vertical change? Step Find the direction. Label the drawing. The angle ou are looking for is formed b the vector and the -ais. Which trigonometric formula will be used? omplete: tan m ta n 1 10 Holt Geometr

13 8 The component form of a vector lists the horizontal and vertical change from the initial point to the terminal point. To reach a campsite, a hiker first walks for 3 miles at a bearing of N 50 E. She then walks 5 miles due east. How far is the hiker from where she started and what direction? Round speed to the nearest tenth and the direction to the nearest degree. Understand the Problem 1. What are ou asked to find? Make a Plan Read To Go On? Problem Solving Intervention 8-6 Vectors. Write the vector in form for the hiker and the resultant. Solve 3. omplete the vector sketches. First Walk N N Second Walk W 50 3 E W 5 E S S. What angle does the vector in the First Walk make with the -ais? Write the vector for the hiker in component form. cos 0, so 3 cos 0. 3 sin 0 3, so sin 0. The hiker s vector is, Write the vector for traveling east in component form., 0 7. Find the resultant vector:, 1.9, 0, Find the magnitude of the resultant vector. (7.3 0 ) ( 0 ) 9. The angle measure formed b the resultant vector gives the actual direction. tan, so ta n or N 75 E. Look ack 10. Plot our findings on the graph. Does our result make sense? 11 Holt Geometr

14 8 Read To Go On? Quiz 8- ngles of Elevation and Depression 1. The scout at the top of a 1800-ft mountain spots a campsite. He measures the angle of depression to be 33. How far is the campsite from the foot of the mountain? Round to the nearest foot m. The angle of elevation from a ship to the top of a lighthouse is. If the ship is 100 m from the lighthouse, how tall is the lighthouse? 100 m 8-5 Laws of Sines and Laws of osines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3.. EG E 3 F 5 57 G G E 7 G 9 8 F Holt Geometr

15 8 Read To Go On? Quiz continued 8-6 Vectors Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 9. 6, 10. 5, 11. 1, Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 1. wind velocit is given b the vector 13. The path of a hiker is given b the 3,. vector 6, The velocit of a plane is given b the vector 5, canoeist leaves shore at a bearing of N 50 E and paddles at a constant speed of 6 mi/h. There is a mi/h current moving due east. What are the canoe s actual speed and direction? Round the distance to the nearest tenth of a mile and the direction to the nearest degree Holt Geometr

16 8 Read To Go On? Enrichment Law of Sines and Law of osines Solve each problem. Sketch a diagram and identif whether ou will use the Law of Sines or Law of osines. 1. To find the distance between two points and on opposite sides of a small pond, a surveor determines that is feet, angle is 56., and angle is Find the distance between. Round to the nearest tenth.. To find the distance XY across a canon the following measurements were taken. XZ is 570 ards, angle YXZ is 103. and angle XZY is 3.6. Z X Y 3. surveor is tring to determine the distance between points and. His view is obstructed b a large barn. He determines that is 75 feet, is 58 feet and angle is 83. Find to the nearest foot.. grove of trees is obstructing the view of a surveing crew. The have determined that the following distances: 13 feet, 15 feet and the measure of angle is 8.. What is the measure of? Round to the nearest foot. 5. boat race starts at point J proceeds to point K and then point L before returning to the starting point. Using the diagram shown, determine the total distance of the race. K J 53 9 km L 1 Holt Geometr