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Unit 2: Right Triangles SMART Packet #4 Identifying sides and angles of a right triangle Student: Teacher: Standards A.A.45 Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides G.G.48 Investigate, justify, and apply the Pythagorean Theorem and its converse The Pythagorean Theorem The Pythagorean Theorem: In a right triangle, if a and b are the measures of the legs, and c is the hypotenuse, then a2 + b2 = c2 c a b Example 1: Find the length of the hypotenuse if a = 9 and b = 12. a2 + b2 = c2 92 + 122 = c2 81 + 144 = c2 225 = c2 Substitute the given values into the Pythagorean Theorem Simplify. Take the square root of both sides 225 c 2 15 = c To find 225 , press: 2nd x2 2 2 5 ENTER PRACTICE, PART I Find the length of the missing side. 1. a = 7, b = 24, c = ? a2 + b2 = c 2 ( )2 + ( )2 = ( )2 2. a = 8, b = 15, c = ? 3. a = 12, b = 5, c = ? 4. 5. Example 2: Find the length of the leg, to the nearest tenth, if a = 4 and c = 10. a2 + b2 = c2 42 + b2 = 102 16 + b2 = 100 –16 –16 b2 = 84 Substitute the given values into the Pythagorean Theorem Solve for b. Take the square root of both sides. b 2 84 b = 9.165... Rounding to the nearest tenth: b = 9.2 9.165... . T E N T H S 6>5 The number to the right of the tenths place is greater than 5, so round 1 up to 2. PRACTICE, PART II Find the length of the missing side. Round decimals to the nearest tenth. 6. a = 3, b = ?, c = 5 7. a = ?, b = 7, c = 10 8. 9. Example 3: A 12-ft ladder is placed 5 ft from a building. How tall is the building, to the nearest tenth? 12 ft 5 ft Substitute the values into the Pythagorean Theorem & solve for the missing side. a2 + b2 = c2 52 + b2 = 122 25 + b2 = 144 b2 = 119 b = 10.9 The building is 10.9 feet tall. PRACTICE, PART III: Regents Prep (with pictures) 10. The NuFone Communications Company must run a telephone line between two poles at opposite ends of a lake, as shown in the diagram. The length and width of the lake are 75 feet and 30 feet, respectively. What is the distance between the two poles, to the nearest foot? (1) 105 (2) 69 (3) 81 (4) 45 11. Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards. What is the length of the diagonal, in yards, that Tanya runs? (1) 50 (2) 70 (3) 60 (4) 80 12. Don placed a ladder against the side of his house as shown below. (a) Find the value of x, the distance from the foot of the ladder to the base of the house (b) Simplify the equations below. Which one gives you the same value you found for x in part (a)? That is your answer! (1) x = 20 – 19.5 2 2 (2) x 20 19.5 (3) x = 202 – 19.52 2 2 (4) x 20 19.5 13. The diagram shows a kite that has been secured to a stake in the ground with a 20-foot string. The kite is located 12 feet from the ground, directly over point X. What is the distance, in feet, between the stake and point X? 14. A wall is supported by a brace 10 feet long, as shown. If one end of the brace is placed 6 feet from the base of the wall, how many feet up the wall does the brace reach? PRACTICE, PART IV: Regents Prep. (without pictures) Draw a picture AND label it with the given information. 15.A woman has a ladder that is 13 feet long. If she sets the base of the ladder on level ground 5 feet from the side of a house, how many feet above the ground will the top of the ladder be when it rests against the house? 16. The length of the hypotenuse of a right triangle is 34 inches and the length of one of its legs is 16 inches. What is the length, in inches, of the other leg of this right triangle? (1) 16 (3) 25 (2) 18 (4) 30 17. An 18-foot ladder leans against the wall of a building. The base of the ladder is 9 feet from the building on level ground. How many feet up the wall, to the nearest tenth of a foot, is the top of the ladder? 18. If the length of a rectangular television screen is 20 inches and its height is 15 inches, what is the length of its diagonal, in inches? (1) 5 (2) 25 (3) 13.2 (4) 35