Unit 3 - Study Guide Answers

Questions 1 & 2 The Pythagorean Theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. The Pythagorean Theorem can be represented by the equation a2 + b2 = c 2, where a and b are the length of the legs of the triangle, and c is the length of the hypotenuse.

Question 3 Use what you know about the Pythagorean Theorem to label the sides of the following triangle.   leg hypotenuse leg

Question 4 Is a triangle with the lengths 7, 24, and 25 a right triangle. Why or why not. Show work to indicate how you got your answer. Use the Pythagorean Theorem to determine if this is a right triangle. 72 + 242 = 252 49 + 576 = 625 625 = 625 The sum of the squares of the legs is equal to the square of the hypotenuse, so it is a right triangle.

Question 5 A football field is 360 feet by 45 feet. How long is the walk from one corner diagonally to the opposite corner? 360 ft 452 + 3602 = c2 2025 + 129600 = c2 √131625 = √c2 362.8 ft = c 45 ft

Question 6 Using the illustration below, what is the approximate height of the hot air balloon? a2 + 13252 = 20002 a2 + 1755625 = 4000000 √a2 = √2244375 a = 1498.1 ft

Question 7 A rectangle has a diagonal 25 inches long and a width of 6 inches. What is the length of the rectangle? 62 + b2 = 252 36 + b2 = 625 √b2 = √589 b = 24.3 in. 25 in 6 in

Question 8 Which of the following measures are valid measures of the sides of a right triangle? Explain your reasoning. A. 3, 4, 7 B. 5, 12, 13 C. 20, 21, 28 D. 12, 37, 34 5, 12, and 13 are valid measures of a right triangle because they form a Pythagorean Triple

Question 9 A spider has taken up residence in a small cardboard box which measures 2 inches by 4 inches by 4 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? 𝒍 𝟐+ 𝒘𝟐+𝒉𝟐 𝟒 𝟐+ 𝟒𝟐+𝟐𝟐 = 𝟏𝟔+𝟏𝟔+𝟒 𝟑𝟔 =𝟔 𝒊𝒏2

Question 10 A package is in the shape of a cube. The height of the package is 10 inches. What is the diagonal length of the package? 𝒍 𝟐 +𝒘𝟐+𝒉𝟐 𝟏𝟎 𝟐+ 𝟏𝟎𝟐+𝟏𝟎𝟐 = 𝟏𝟎𝟎+𝟏𝟎𝟎+𝟏𝟎𝟎 𝟑𝟎𝟎 =𝟏𝟕.𝟑 𝒊𝒏2 10 in

Question 11 Find the length of the missing side. a2 + b2 = c2 5.8 in = c 3 in 5 in

Questions 12 & 13 Solve for y: 2 𝑦 3 = 16 y3 = 8 y = 3 8 y = 2 Solve for z: 3 𝑧 2 = 108 𝑧 2 = 36 z = 36 Z = ±6

Questions 14, 15, & 16 Explain what the word volume means. The measure of the space occupied by a solid How does the volume of a cylinder compare to the volume of a cone? The volume of a cylinder is three times greater than the volume of a cone How does the volume of a cylinder and cone compare to the volume of a sphere? The volume of a sphere is double the volume of a cone and 2/3rd the volume of a cylinder.

Question 17 A candle maker uses a cylinder mold, which is 18 inches tall and has a radius of 1 inch. What is the volume of the candle mold? r = 1 in. 𝑽=𝝅 𝒓 𝟐 𝒉 𝑽=𝝅 𝟏 𝟐 •𝟏𝟖 V = 18 𝝅 V = 56.52 in3 18 in.

Question 18 A party hat is in the shape of a cone with a radius of 3 in. and a height of 5 in. What is the volume of the party hat? r = 3 in. 𝑽=𝝅 𝒓 𝟐 𝒉 3 𝑽=𝝅 𝟑 𝟐 •𝟓 V = 15 𝝅 and 47.1 in3 5 in.

Question 19 What is the volume of a beach ball with a radius of 12 centimeters? r = 12 cm 𝑽=𝟒𝝅 𝒓 𝟑 3 𝑽=𝟒𝝅 𝟏𝟐 𝟑 V = 2304 𝝅 and 7234.6 cm3

Question 20 What is the volume of the following cone? 𝑽=𝝅 𝒓 𝟐 𝒉 3 𝑽=𝝅 𝟓 𝟐 •𝟓 V = 41.6 𝝅 and 130.8 mm3

Question 21 Find the volume of the sphere shown below. 𝑽=𝟒𝝅 𝒓 𝟑 3 𝑽=𝟒𝝅 𝟐 𝟑 V = 10.6 𝝅 and 33.5 in3

Question 22 Find the volume of the cylinder below. 𝑽=𝝅 𝒓 𝟐 𝒉 𝑽=𝝅 𝟐.𝟓 𝟐 •𝟕 V = 137.4 in3

Question 22 Find the volume of the cylinder below in terms of pi. 𝑽=𝝅 𝒓 𝟐 𝒉 𝑽=𝝅 𝟐.𝟓 𝟐 •𝟕 V = 𝟔.𝟐𝟓•𝟕𝝅 V = 43.75 𝝅 in3 If you are finding the volume “in terms of pi”, DO NOT multiply by 3.14.

Question 23 The volume of a cylinder is 12.56 𝑚 3 . If the height of the cylinder is 1 m, what is its diameter? 𝑽=𝝅 𝒓 𝟐 𝒉 𝟏𝟐.𝟓𝟔=𝝅 𝒓 𝟐 •1 Multiply 3.14 by 1 Divide both sides by 3.14 to isolate r 2 4 = r2 Find the square root of 4 r = 2 m So the diameter is 4 m r = ? m h = 1 m

Question 24 What are the two methods of finding the distance between two points? Given two points, you can always plot them, draw the right triangle, and then find the length of the hypotenuse. The length of the hypotenuse is the distance between the two points. Another way to find the distance between two points is algebraically with a formula.

Question 25 What is the distance between P1 and P2? The legs are 6 and 9, now I can find the distance by finding the hypotenuse of the triangle: a2 + b2 = c2 62 + 92 = c2 36 + 81 = c2 √117 = √c2 10.8 = c 6 units 9 units

Question 26 What is the distance between P1 and P2? The legs are 3 and 4, now I can find the distance by finding the hypotenuse of the triangle: a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 √25 = √c2 5 = c 3 units 4 units

Question 27 Given points C(-6, 10) and D(-3, -2), what is the length of CD? I can use the distance formula: d = 𝒙𝟐−𝒙𝟏 𝟐+ 𝒚𝟐 −𝒚𝟏 𝟐 d = −𝟑−−𝟔 𝟐+ −𝟐 −𝟏𝟎 𝟐  d = 𝟑 𝟐+ −𝟏𝟐 𝟐 d = 𝟗+𝟏𝟒𝟒  d = 𝟏𝟓𝟑 CD = 12.4

Question 28 Given points S(-4, -2) and T(-1, 0), what is the length of ST? I can use the distance formula: d = 𝒙𝟐−𝒙𝟏 𝟐+ 𝒚𝟐 −𝒚𝟏 𝟐 d = −𝟏−−𝟒 𝟐+ 𝟎 −−𝟐 𝟐  d = 𝟑 𝟐+ 𝟐 𝟐 d = 𝟗+𝟒  d = 𝟏𝟑 ST = 3.6