Presentation on theme: "Problem Solving with Right Triangles"— Presentation transcript:
1 Problem Solving with Right Triangles Chapter 12 - Lesson 12.2Problem Solving with Right TrianglesHW: 12.2/1-20
2 Finding the Sides of a Triangle Remember: SOHCAHTOA
3 First we will find the Sine, Cosine and Tangent Review: Trig RatiosFirst we will find the Sine, Cosine and Tangentratios for Angle P.PNext we will find the Sine, Cosine, and Tangent ratios for Angle Q2012AdjacentQ16OppositeRemember SohCahToa
4 Solving Right Triangles Every right triangle has one right angle, two acute angles, one hypotenuse, and two legs.To Solve a Right Triangle means to determine the measures of all six parts.
9 What are the angles of elevation and depression and what is their relationship to right triangles?
10 Looking down from the horizontal ANGLE OF DEPRESSIONLooking down from the horizontalobserverEye levelAngle of depressioncliffsobjectSea level
11 Looking up from the horizontal ANGLE OF ELEVATIONLooking up from the horizontalobserverEye levelcliffsAngle of elevationobjectSea level
12 If an observer sights an object above, the angle between a horizontal line and his or her line of sight is called an angle of elevation. If the line of sight is below the horizontal it is called the angle of depression.Angle of DepressionAngle of Elevation
15 they are alternate interior angles The angles are equal –they are alternate interior angleseye - levelline of sighteye - level
16 Angles of Elevation and Depression Top HorizontalAngle of DepressionLine of SightAngle of ElevationBottom HorizontalSince the two horizontal lines are parallel, by Alternate Interior Angles the angle of depression must be equal to the angle of elevation.
17 Angles of Elevation and Depression line of sight
18 Step 1: Draw a right angled triangle with the given information. Example 1The angle of elevation of building A to building B is The distance between the buildings is 21 meters. Calculate how much taller Building B is than building A.Step 1: Draw a right angled triangle with the given information.Angle of elevationBA21 mh250Step 2: Take care with placement of the angle of elevationStep 3: Set up the trig equation.Step 4: Solve the trig equation.
19 Step 1: Draw a right angled triangle with the given information. Example 2A boat is 60 meters out to sea. Madge is standing on a cliff 80 meters high. What is the angle of depression from the top of the cliff to the boat?Step 1: Draw a right angled triangle with the given information.60 m80 mAngle of depressionStep 2: Alternate interior angles place inside the triangle.Step 3: Decide which trig ratio to use.Step 4: Use calculator to find the value of the unknown.
20 Step 1: Draw a right angled triangle with the given information. Plane Example 3Marty is standing on level ground when he sees a plane directly overhead. The angle of elevation of the plane after it has travelled 25 km is Calculate the altitude of the plane at this time.Step 1: Draw a right angled triangle with the given information.Planeh25 km200Step 2: Alternate interior angles places 200 inside the triangle.Angle of elevation200Step 3: Decide which trig ratio to use.Step 4: Use calculator to find the value of the unknown.(nearest km)
22 Therefore, Petra is closer to the tree, since the distance is shorter. Example 4Kate and Petra are on opposite sides of a tree. The angle of elevation to the top of the tree from Kate is 45o and from Petra is 65o. If the tree is 5 m tall, who is closer to the tree, Kate or Petra?KP4506505mkpKatePetraAnswerTherefore, Petra is closer to the tree, since the distance is shorter.
23 The bird is 6.6 m (2 + 4.6) from the ground at that moment. Example 5Maryann is peering outside her window. From her window she sees her car and a bird hovering above her car. The angle of depression of Maryann’s car is 200 whilst the angle of elevation to the bird is If Maryann’s window is 2m off the ground, what is the bird’s altitude at that moment?Step 1: Draw a diagramStep 2: Set up the trig equations in two parts. Find d first, then b.BirdCar4002002 mStep 3: Solve the equations and answer the question.bdTherefore,The bird is 6.6 m ( ) from the ground at that moment.
24 Your Turn 1:You sight a rock climber on a cliff at a 32o angle of elevation. The horizontal ground distance to the cliff is 1000 ft. Find the line of sight distance to the rock climber.x1000 ft
25 Your Turn 2:An airplane pilots sights a life raft at a 26o angle of depression. The airplane’s altitude is 3 km. What is the airplane’s surface distance d from the raft?3 kmd
26 FYI: TheodoliteTheodolite is a precision instrument for measuring angles in the horizontal and vertical planes.Theodolites are mainly used for surveying applications, and have been adapted for specialized purposes in fields like meteorology and rocket launch technology.When the telescope is pointed at a target object, the angle of each of these axes can be measured with great precisionTheodolites are still used today for ultra high precision optical alignment and measurement
27 Your Turn 3:A surveyor stands 200 ft from a building to measure its height with a 5-ft tall theodolite. The angle of elevation to the top of the building is 35°. How tall is the building?x = 200 • tan 35°tan 35° =x200So x ≈ 140To find the height of the building, add the height of the Theodolite, which is 5 ft tall.The building is about 140 ft + 5 ft, or 145 ft tall.
28 Your Turn 4: x miles ≈ 18.9939 miles ≈ 19 miles An airplane flying 3500 ft above ground begins a 2° descent to land at an airport. How many miles from the airport is the airplane when it starts its descent?sin 2° =3500x ftx ft =3500 ftsin 2°x ft ≈ 100, ftx miles ≈ miles ≈ 19 milesThe airplane is about 19 mi from the airport when it starts its descent.
29 Your Turn 5:A 6-ft man stands 12 ft from the base of a tree. The angle of elevation from his eyes to the top of the tree is 40°.1. About how tall is the tree?2. If the man releases a pigeon that flies directly to the top of the tree, about how far will it fly?3. What is the angle of depression from the treetop to the man’s eyes?about 16 ftabout 15.7 ft40°
30 Your Turn 6:CIRCUS ACTS. At the circus, a person in the audience watches the high-wire routine. A 5-foot-6-inch tall acrobat is standing on a platform that is 25 feet off the ground. How far is the audience member from the base of the platform, if the angle of elevation from the audience member’s line of sight to the top of the acrobat is 27°?27°Step 1: Draw a triangle to fit problemx30.5 =Step 2: Label sides from angle’s viewadjoppStep 3: Identify trig function to useS O / HC A / HT O / AStep 4: Set up equation30.5tan 27° =xStep 5: Solve for variablex tan 27° = 30.5x = (30.5) / (tan 27°)x = 59.9
31 Your Turn 7:DIVING At a diving competition, a 6-foot-tall diver stands atop the 32-foot platform. The front edge of the platform projects 5 feet beyond the ends of the pool. The pool itself is 50 feet in length. A camera is set up at the opposite end of the pool even with the pool’s edge. If the camera is angled so that its line of sight extends to the top of the diver’s head, what is the camera’s angle of elevation to the nearest degree?37 =45 =x°
32 Your Turn 8:From a point 80m from the base of a tower, the angle of elevation is 28˚. How tall is the tower to the nearest meter?x28˚80mtan 28˚ =80 (tan 28˚) = x80 (.5317) = xx ≈ 42.5m is the height of the tower
33 Your Turn 8:A ladder that is 20 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75˚, how far is the bottom of the ladder from the base of the building?20buildingladder75˚xcos 75˚ =20 (cos 75˚) = x20 (.2588) = xx ≈ 5.2ft from the base of the building
34 Your Turn 9:When the sun is 62˚ above the horizon, a building casts a shadow 18m long. How tall is the building?x62˚18shadowtan 62˚ =18 (tan 62˚) = x18 (1.8807) = xx ≈ 33.9m is the height of the building
35 Your Turn 10:A kite is flying at an angle of elevation of about 55˚. Ignoring the sag in the string, find the height of the kite if 85m of string have been let out.kite85xstring55˚sin 55˚ =85 (sin 55˚) = x85 (.8192) = xx ≈ 69.6m is the height of the kite
36 Your Turn 11:A 5.5 foot person standing 10 feet from a street light casts a 14 foot shadow. What is the height of the streetlight?5.5x˚1014shadow1st find the angle of elevation2nd use the angle to find the height of the light.tan x˚ =x° ≈ °height = 9.43 feet
37 Your Turn 12:The angle of depression from the top of a tower to a boulder on the ground is 38º. If the tower is 25m high, how far from the base of the tower is the boulder?38ºangle of depression25Alternate Interior Angles are congruent38ºx