Basic Lesson: Pythagorean Theorem
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1 Basic Lesson: Pythagorean Theorem Basic skill One leg of a triangle is 10 cm and other leg is of 24 cm. Find out the hypotenuse? Here we have AB = 10 and BC = 24 Using the Pythagorean Theorem AC 2 = AB 2 + BC 2 Replacing value AB with 10 and BC with 24 in above formula we have AC 2 = AB 2 + BC 2 (x) 2 = (10) 2 + (24) 2 x= 676 = 26 Answer = 26 cm Basic skills Practice Find the length of a rod that has to be fixed diagonally in a room of dimensions 24 feet by 28 feet by 30 feet. Rod has to be fixed in a room diagonally which basically means diagonal of a cube. Diagonal of a cube = (l 2 +b 2 +h 2 ) Replacing values in above formula, Diagonal = ( ) = = 2260 = feet Answer = feet
2 Intermediate Lesson: Pythagorean Theorem Intermediate skills Practice A square with sides of 40 feet. What is the shortest distance between two opposite vertices? Here we have AB = 40 and BC = 40 Using the Pythagorean Theorem AC 2 = AB 2 + BC 2 Replacing value AB and BC with 40 in above formula we have AC 2 = AB 2 + BC 2 (40) 2 + (40) 2 = (x) 2 (x) 2 = = 3200 x= 3200 = (The diagonal of a square can also be calculated by using formula a 2, where a is the side of the square. Applying this formula for this problem; diagonal=40 x 2 = 40 x = In each case, answer is same) Answer = feet Intermediate skill To avoid the pond, Joe must walk 14 meters south and 48 meters east. To the nearest meter, how many meters would be saved if it were possible to walk through the pond? Suppose we have AC = 14 and AB = 48 If it were possible to walk through the pond, shortest distance could be calculated by using the Pythagorean Theorem AC 2 + AB 2 = BC 2 Replacing value AC with 14 and AB with 48 in the formula we have (14) 2 + (48) 2 = (BC) 2 (BC) 2 = = 2500 BC= 2500 = 50 Since, it is not possible to walk through the pond, Joe must walk 14m +48m = 62 Distance that would be saved if it were possible to walk through the pond = total distance-shortest distance= 62-50=12 Answer = 12 m
3 Independent Practice 1: Pythagorean Theorem 1. A rectangle has a width of 6 feet and a length of 8 feet. Find the length of the diagonal in feet. 2. A rectangle has a width of 14 inches and a diagonal of 50 inches. Find the length of the rectangle in inches. 3. A 65 foot ladder is leaned against a wall. If the base of the ladder is 63 feet from the wall, how high up the wall will the ladder reach? Firefighters have a 37 feet extension ladder. In order to reach 35 feet up a building, how far away from the building should the feet of the ladder be placed? Princess Marie is locked in the tower of a castle. Tim volunteered to rescue the princess. If the tower window is 480 feet above the ground and you must place your ladder 31 feet from the base of the castle (because of the moat), what is the shortest length ladder, to the nearest foot, you will need to reach the princess? Tom wants to swim across a river that is 400 meters wide. He begins swimming perpendicular to the shore he started from but ends up 300 meters down river from where he started because of the current. How far did he actually swim from his starting point? In the South, settlers often fashioned tents as an isosceles triangle. How long would the cloth have to be so that the opening of the tent was 6 meters high and 10 meters wide? 8. A baseball diamond is a square with sides of 80 feet. What is the shortest distance, to the nearest tenth of a foot, between home plate and second base? 9. To avoid the pond, Alex must walk 12 meters south and 35 meters east. To the nearest meter, how many meters would be saved if it were possible to walk through the pond? 10. A square with sides of 50 feet. What is the shortest distance between two opposite vertices? 11. Box measures 32 inches long, 6 inches wide and 8 inches high. What is the diagonal length of the box?
4 12. The sides of a triangle measure 14, 48, and 30. Is this triangle a right triangle? 13. The older floppy diskettes measured 7 and 3 inches on each side. What was the diagonal length of the diskette? 14. A right triangle has a hypotenuse of 17 and a leg of 15. Find the other leg of the triangle? 15. Two joggers run 12 miles north and then 8 miles west. What is the shortest distance to reach their starting point? 16. Slanted sides of a tent 34 feet long total and the bottom of the house is 30 feet across. What is the tallest point of this tent? 17. An equilateral triangle is plotted on a coordinate plane. Two of the vertices are (0,0) and (6,0). Which of the coordinates shown could be the vertex of the third side? 18. One leg of a right triangle is 62 units longer than the length of the other leg. If the hypotenuse is 82, then find the other two legs. 19. A cube has a width of 4 feet, height of 12 feet and a length of 8 feet. Find the length of the diagonal in feet. 20. A 15 foot ladder is leaned against a wall. If the base of the ladder is 9 feet from the wall, how high up the wall will the ladder reach?
5 Independent Practice 2: Pythagorean Theorem Sam regularly takes a short-cut across Mr. Hilton's lawn instead of walking on the sidewalk (7 m by 24 m) on his way home from school. How much distance is saved by Joe cutting across the lawn? Tom has covered a distance of 35 meters south and 7 meters east. What can be shortest distance can he walk? Find the length of a diagonal of a cube that has sides measuring 12 cm each. An 11-feet pole casts a shadow of 60 feet. What is the distance between the end of the shadow and the top of pole? A spider has taken up residence in a small cardboard box which measures 3 inches by 7 inches by 5 inches. What is the length, in inches, of a straight spider web that will carry the spider from the lower right front corner of the box to the upper left back corner of the box? David rides his bike 24 km south and then 4 km west. How far is he from his starting point? Town A is 9 miles from town B, and 12 miles from town C. Town A, B and C are forming a right triangle at A. A road connects towns B and C directly. Find the length of this road. A garden is in the shape of a right triangle. It has one side that is 22 ft long, and has a hypotenuse of 122 ft, what is the width of the garden Jack's TV screen is 12 inches long. If the diagonal measures 37 inches, how long is the width of Jack's TV? The foot of a ladder is placed 14 feet from a wall. If the top of the ladder rests 9 feet up on the wall, how long is the ladder? If each of the legs of an isosceles right triangle is 12 inches long, approximate the length of the hypotenuse to the nearest whole number. 12. Find the length of the diagonal of a square whose sides is 16 meters.
6 A hot-air balloon is held in place by the ground crew at a point that is 40 ft from a point directly beneath the balloon. If the rope is of length 41 ft, how far above ground level is the balloon? There is a building with an 11 ft high window. Jack wants to use a ladder to go up to the window, and he decides to keep the ladder 60 ft away from the building to have a good slant. How long should the ladder be? If a leg of a triangle is 10 ft long, and another leg is 8 ft long, what is the length of the hypotenuse? 16. Diagonal of a cube is m. Find the length of the cube Nancy drives her car 15 km south and then 16 km west. How far is she from her starting point? Find x A cartoon in shape of a cube measures 4 inches. What is the diagonal length of the box? If the sum of the sides of a right triangle is 194 inches and the hypotenuse is 170 inches, find the two sides.
7 Homework: Pythagorean Theorem Basic skills Practice Find the length of a rod that has to be fixed diagonally in a room of dimensions 24 feet by 28 feet by 30 feet. Rod has to be fixed in a room diagonally which basically means diagonal of a cube Diagonal of a cube = (l 2 +b 2 +h 2 ) Replacing values in above formula, Diagonal = ( ) = = 2260 = feet Answer = feet 1. A cube has a side of 14 cm. How long is the diagonal? A 145 feet ladder is leaned against a wall. If the base of the ladder is 24 feet from the wall, how high up the wall will the ladder reach If each of the leg of an isosceles right triangle is 20 m long, approximate the length of the hypotenuse to the nearest whole number. Find the length of the diagonal of a square whose sides is 38 meters. AB=12x, AC=5x, BC=65. Find the value of x House A is 38 miles from house B, and 360 miles from house C. House A, B and C are forming a right triangle at A. A road connects houses B and C directly. Find the length of this road. There is a building with a 120 ft high window. Lisa wants to use a ladder to go up to the window, and she decides to keep the ladder 22 ft away from the building to have a good slant. How long should the ladder be? A ground is in the shape of a right triangle. It has one side that is 31 ft long, and has a hypotenuse of 481 ft. 9. Daisy Duck has a nest on the edge of the pond. From her favorite feeding spot, she can either waddle on land around the pond to the nest (70 meters by 240 meters), or she can swim across the pond to the nest. Daisy waddles more quickly than she swims. She waddles at the rate of 25 m/min and she swims at the rate of 10 m/min. Which route is quicker to travel from the feeding spot to the nest? Waddling on land or swimming in the pond?
8 If a leg of a triangle is 30 ft long, and another leg is 224 ft long, what is the length of the hypotenuse? A hall s screen is 62 inches long. If the diagonal measures 80 inches, and length of hall is 42 inches how long is the width of hall s screen? A cartoon measures 12 inches long, 8 inches wide and 18 inches high. What is the diagonal length of the box?
9 Quiz: Pythagorean Theorem Lisa drives her car 48 km south and then 575 km east. How far is she from her starting point? Using the Pythagorean Theorem, find the area of an equilateral triangle whose side measures 4 units. Find the area to the nearest tenth of a square unit. If the legs of an isosceles right triangle are 12 inches long, approximate the length of the hypotenuse to the nearest whole number Dick rides his bike 64 km south and then 1023 km west. How far is he from his starting point? If a leg of a triangle is 23 ft long, and another leg is 264 ft long, what is the length of the hypotenuse? A garden is in the shape of a square of sides 22 feet. What is its hypotenuse? If a side of a triangle is 16 ft long, and another side is 63 ft long, what is the length of the hypotenuse? Town A is 8 miles from town B, and 15 miles from town C. Town A, B and C are forming a right triangle at A. A road connects towns B and C directly. Find the length of this road. Find the height of an equilateral triangle whose side measures 48 cm. 10. A cartoon measures 30 inches long, 4 inches wide and 12 inches high. What is the diagonal length of the box? Circle # Correct Percentage Score 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
10 Answer Keys Page 2 - Independent Practice: 1 10 feet 2 48 inches 3 16 feet 4 12 feet feet meters meters feet feet inches 12 no inches miles feet 17 3, , feet feet Page 3 - Independent Practice: 1 5 meters meters feet inches km 7 15 miles feet 9 35 inches inches meters 13 9 feet feet feet meters km inches 20 26, 168 inches Page 4 - Home Work : cm feet m meters miles feet feet 9 waddle feet inches inches Page 5 - Quiz : km units inches km feet feet 7 65 feet 8 17 miles cm inches
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