Lesson 3: Using the Pythagorean Theorem The Pythagorean Theorem only applies to triangles. The Pythagorean Theorem + = Example 1 A sailboat leaves dock and travels 6 mi due east. Then it turns 90 degrees and sails 4 miles due south. At this point, how far is it from the dock?
Example 2 Find the length of the hypotenuse of the right triangle. 13 6 Find the length of the missing leg. Example 3 61 11
Lesson 3: Using the Pythagorean Theorem Part 1: Find the missing leg or hypotenuse using the Pythagorean Theorem. (All of the triangles below are right triangles.) Round your answers to the nearest hundredth. 1. 5. 22.6 15 13 20 2. 3 6. 8 22 33 3. 16 3 7. 13.5 38 4. 12 8. 35 72.25 40
Part 2: Determine whether the following triangles are right triangles. 1. 7 2. 24 25 10 19 14.5 3. 40 9 4. 12 8 41 5
Part 3: Applying Your Knowledge 1. John leans a 16 foot ladder against the side of his house in order to clean the gutters. The side of the house is 15 feet high. How far out from the house is the bottom of the ladder? 2. You leave home and walk 7 blocks north. You turn and walk 5 more blocks west in order to reach school. How far have you walked to school? How far will you walk if you take a direct (straight line) route to your school? Which route is shorter? How much shorter is it? Explain how you determined your answer. 3. A wire is run from one cell tower to another cell tower. If the towers are 50 meters apart, how much wire is needed? 10m 55m 4. In the trapezoid below, find the missing side. 25 cm 29 cm 32 cm 1. Determine whether or not the following triangle is a right triangle. Explain your answer. (3 points) (Give or take a few tenths for rounding purposes) Leg 1: 16 Leg 2: 18.5 Hypotenuse: 24.46 2. Jessie biked 8 miles north, 12 miles east and then 9 miles north again. (4 points) How far did Jessie bike? If Jessie had traveled in a straight line from his starting point to his end point, how many fewer miles would he have biked? Explain how you determined your answer.
Lesson 3: Using the Pythagorean Theorem Answer Key Part 1: Find the missing leg or hypotenuse using the Pythagorean Theorem. (All of the triangles below are right triangles.) Round your answers to the nearest hundredth. 1. 15 20 The hypotenuse is missing (c) 15 2 + 20 2 = c 2 Substitute 225+400 = c 2 625 = c 2 625= 25 = c The hypotenuse is 25. 5. 13 22.6 A leg is missing (a) a 2 +13 2 = (22.6) 2 a 2 + 169= 510.76 a 2 + 169-169 =510.76-169 a 2 = 341.76 = 341.76 a = 18.49 The missing leg is 18.49. 2. 3. 4. 8 3 12 16 3 A leg is missing (a) a 2 + 3 2 = 8 2 Substitute a 2 + 9 = 64 a 2 + 9-9 =64-9 a 2 = 55 = 55 a = 7.42 The missing leg is 7.42. The hypotenuse is missing (c) 16 2 + 3 2 = c 2 Substitute 256+9 = c 2 265 = c 2 265= 16.28 = c The hypotenuse is 16.28. 72.25 The hypotenuse is missing (c) 12 2 + ( 72.25) 2 = c 2 144+72.25 = c 2 216.25= c 2 216.25= 14.71 = c The hypotenuse is 14.71. 6. 22 33 7. 38 13.5 8. 35 40 A leg is missing (a) a 2 +22 2 = 33 2 a 2 + 484= 1089 a 2 + 484-484 =1089-484 a 2 = 605 = 605 a = 24.6 The missing leg is 24.6. A leg is missing (a) a 2 +( 38) 2 =13.5 2 a 2 + 38= 182.25 a 2 + 38-38 =182.25-38 a 2 = 144.25 = 144.25 a = 12.01 The missing leg is 12.01. A leg is missing (a) a 2 +35 2 =40 2 a 2 + 1225= 1600 a 2 + 1225-1225 =1600-1225 a 2 = 375 = 375 a = 19.36 The missing leg is 19.36.
Part 2: Determine whether the following triangles are right triangles. 1. 7 2. 24 25 10 19 14.5 We can use the Pythagorean Theorem to determine if the triangle is a right triangle. Step 1: Substitute the numbers into the formula: a 2 + b 2 = c 2 7 2 + 24 2 = 25 2 49+ 576 = 625 625=625 Yes, since this works perfectly, this triangle is a right triangle. We can use the Pythagorean Theorem to determine if the triangle is a right triangle. Step 1: Substitute the numbers into the formula: a 2 + b 2 = c 2 10 2 + 14.5 2 = 19 2 100+ 210.25 = 361 310.25 = 361 No, 310.25 361; therefore, this is not a right triangle. 3. 40 9 4. 12 8 41 We can use the Pythagorean Theorem to determine if the triangle is a right triangle. Step 1: Substitute the numbers into the formula: a 2 + b 2 = c 2 9 2 + 40 2 = 41 2 81 + 1600 = 1681 1681=1681 Yes, since this works perfectly, this triangle is a right triangle. 5 We can use the Pythagorean Theorem to determine if the triangle is a right triangle. Step 1: Substitute the numbers into the formula: a 2 + b 2 = c 2 5 2 + 8 2 = 12 2 25+64 = 144 89=144 No, 89 144; therefore, this triangle is not a right triangle.
Part 3: Applying Your Knowledge 1. John leans a 16 foot ladder against the side of his house in order to clean the gutters. The side of the house is 15 feet high. How far out from the house is the bottom of the ladder? Ladder If you draw a picture, you get a better idea of what the right triangle might look like in order to solve this problem. A side is missing, therefore, we will use the Pythagorean Theorem to solve for the missing side. 16 a 15 a 2 + b 2 = c 2 a 2 + 15 2 = 16 2 a 2 + 225 = 256 a 2 + 225 225 = 256-225 a 2 = 31 = 31 a = 5.57 The bottom of the ladder is placed 5.57 feet away from the house. 2. You leave home and walk 7 blocks north. You turn and walk 5 more blocks west in order to reach school. How far have you walked to school? 12 blocks ( 7+5 = 12) How far will you walk if you take a direct (straight line) route to your school? Which route is shorter? How much shorter is it? Explain how you determined your answer. 5 blocks Drawing a picture will help you to visualize the right triangle. The direct route is the hypotenuse of the right triangle. Direct route 7 blocks a 2 + b 2 = c 2 5 2 + 7 2 = c 2 25 + 49 = c 2 74 = c 2 74= 8.6 = c The direct route is shorter by 3.4 blocks. I found this answer by using the Pythagorean Theorem to find the distance of the direct route. (8.6 blocks) Then I subtracted 8.6 from the actual distance (12 blocks) and found an difference of 3.4 blocks. The direct route would be 8.6 blocks long.
3. A wire is run from one cell tower to another cell tower. If the towers are 50 meters apart, how much wire is needed? First create the right triangle. 45 (55-10) The distance between the 10m 55m towers is 50 and if you make 50 m the base of the triangle at the top of the shortest tower, a 2 + b 2 + c 2 then we have to figure out the height of the other leg. 50 2 + 45 2 = c 2 Substitute for a and b. (55-10 = 45) We will be 2500 + 2025 = c 2 solving for the hypotenuse. 4525 = c 2 4525= 67.27 = c 67.27 meters of wire is needed to run the wire from one cell tower to another cell tower. 4. In the trapezoid below, find the missing side. 29 cm 25 cm 32 cm The missing side is the hypotenuse of the right triangle that can be formed. We know the height of the triangle is 29 cm since the width of the rectangle is 29 cm. However, we must find the length of the base of the triangle. Since the entire length of the trapezoid is 32 cm and the length of the rectangle is 25 cm, we can subtract: 32-25 = 7 cm. The base of the triangle is 7 cm. We know one leg is 29cm and one leg is 7cm. We must find the hypotenuse. a 2 + b 2 = c 2 29 2 + 7 2 = c 2 841 + 49 = c 2 890 = c 2 890= 29.83 = c The missing side of the trapezoid is about 29.83 cm long.
1. Determine whether or not the following triangle is a right triangle. Explain your answer. (3 points) (Give or take a few tenths for rounding purposes) Leg 1: 16 Leg 2: 18.5 Hypotenuse: 24.46 We will use the Pythagorean Theorem to determine whether or not this is a right triangle. Pythagorean Theorem 16 2 + 18.5 2 = 24.46 2 Substitute the leg values for A & B and the hypotenuse for C 256 + 342.25 = 598.29 Evaluate 598.25 = 598.29 (Due to rounding, we can say that these two numbers are equal since they are only off by 4 hundredths. ) This is a right triangle because the sum of the squares of the legs is equal to the square of the hypotenuse. If then we know that the triangle is a right triangle.
2. Jessie biked 8 miles north, 12 miles east and then 9 miles north again. (4 points) How far did Jessie bike? 29 miles (8 + 12 + 9 = 29 miles) If Jessie had traveled in a straight line from his starting point to his end point, how many fewer miles would he have biked? Explain how you determined your answer. 9 17 First draw Jessie s bike path. 8 miles north, 12 east, and 9 north. (These lines are in red). Then I drew in the straight line to represent the direct route from beginning to end. (This is the blue solid line). This is the distance that I need to find, therefore, I need to create a right triangle with this being the hypotenuse. 8 START a 2 + b 2 = c 2 12 2 + 17 2 = c 2 144 + 289 = c 2 433 = c 2 12 12 8 If I draw the base of the triangle (dotted blue line), I know this distance is 12 because it s the same distance as Jessie traveling east. Then if I draw the a line up to meet the red line, I know this distance is 8 since it s the same as Jessie traveling north 8 miles. That means that the other leg of the triangle is 17. (8+9 = 17) 433= 20.81 = c If Jessie traveled in a straight line this distance would be 20.81 miles. Therefore, he would have biked 8.19 fewer miles. I subtracted 29 20.81 to get a difference of 8.19. I have one leg as 12, another leg as 17 and I am solving for the hypotenuse.