1. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Solving Right Triangles Section 3.3

2. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Solve a Right Triangle A city at point C, on the other side of the mountain, needs the water. Engineers will build a tunnel through the mountain, from C to S. How can they determine the direction and the distance of the tunnel? An island has a spring at point S, on one side of a mountain, M. The Tunnel of Samos N

3. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. The distance from C to S cannot be measured directly. Instead, engineers follow a path around the mountain at a constant elevation, shown in red. Solve a Right Triangle The Tunnel of Samos N

4. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. As they move along the path, the engineers keep track of how far they have moved west or east, and how far they have moved north. They use two rods of adjustable length that are set perpendicular to each other. One rod points west, while the other points north. The ends of the rods form a right angle. The rods are used as often as needed on the way around the mountain. Solve a Right Triangle The Tunnel of Samos

5. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. The right-angle figures measured by the rods are known as transverses. For simplicity, only four traverses are shown in the diagram. On the real island, there would be hundreds or thousands of traverses. Solve a Right Triangle The Tunnel of Samos N

6. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Suppose that the distance west measured by the transverses is 984 m, the distance north is 946 m, and the distance east, measured around the other side of the mountain, is 563 m. These distances are shown in red. Solve a Right Triangle The Tunnel of Samos Model the actual distance of the path using a right triangle by subtracting the distance east from the distance west. This is the base of the triangle, shown in green. The height of the triangle is the distance north. Total west distance = west – east = 984 – 563 = 421

7. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Use the Pythagorean theorem to determine hypotenuse, c, which is the length of the tunnel. Solve a Right Triangle The Tunnel of Samos The length of the tunnel is 1035 m. c 2 = a 2 + b 2 c 2 = 421 2 + 946 2 c= 1035 S C

8. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Use the tangent ratio to find the angle at C, the direction of the tunnel. Solve a Right Triangle The Tunnel of Samos The direction of the tunnel is 66°north of west. S

9. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Historical Notes This tunnel was built on the Greek island of Samos, near the Turkish coast, in the sixth century B.C. Pythagoras lived on Samos. Although no one is sure how the builders determined the direction, it is likely they used a method similar to the one described here. Solve a Right Triangle The Tunnel of Samos They worked without magnetic compasses, optical instruments, or topographical maps. The tunnel remains, and can be visited.

10. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. The town of Moshi, Tanzania has an elevation of 900 m above sea level. The highest peak of Mt. Kilimanjaro, Kibo is north of the town at an angle of elevation of 9.7°. From a point 10.0 km east of Moshi, the peak of the mountain is 71.1° north of west. What is the height of Kibo peak above sea level? Solve a Double Triangle Problem The Height of Mt. Kilimanjaro Kibo Peak

11. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Start with a triangle to find the distance from Moshi to the peak, d. Solve a Double Triangle Problem The Height of Mt. Kilimanjaro The distance to the peak is approximately 29.2 km. Moshi Peak

12. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Draw a triangle to find the height of the peak, h. Solve a Double Triangle Problem The Height of Mt. Kilimanjaro The height of the peak is approximately 5.0 km or 4991 m. Moshi Peak

13. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Solve a Double Triangle Problem The Height of Mt. Kilimanjaro The height of the peak is approximately 5.0 km or 4991 m. Add the elevation of Moshi to determine the height of the peak above sea level. height of peak + elevation of Moshi = height of peak above sea level 4991 + 900 = height of peak above sea level 5891 = height of peak above sea level The peak of Mt. Kilimanjaro is 5891 m above sea level.

14. Copyright © 2010 McGraw-Hill Ryerson Limited, a Subsidiary of The McGraw-Hill Companies. All rights reserved. Geographical Notes Mt. Kilimanjaro is the highest free-standing mountain in the world. It is an extinct volcano that once had three peaks. Only two remain: Kibo, the highest peak, and Mawenzi. Solve a Double Triangle Problem The Height of Mt. Kilimanjaro