1-6 Two-Dimensional Figures
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1 33. DISC GOLF The diameter of the most popular brand of flying disc used in disc golf measures between 8 and 10 inches. Find the range of possible circumferences and areas for these flying discs to the nearest tenth. The circumference is minimized when the diameter is 8 inches. The minimum circumference is about 25.1 in. The circumference is maximized when the diameter is 10 in. The maximum circumference is about 31.4 in. The area is minimum when the radius is 4 inches. The minimum area is about 50.3 in 2. The area is a maximum when the radius is 5 in. The maximum area is about 78.5 in 2. esolutions Manual - Powered by Cognero Page 1
2 ALGEBRA Find the perimeter or circumference for each figure described. 34. The area of a square is 36 square units. Find the length of the side. Use the formula for the area of a square with side s. Since the length can never be negative, s = 6. Use the formula for perimeter of the square with side s. The perimeter of the square is 24 units. esolutions Manual - Powered by Cognero Page 2
3 35. The length of a rectangle is half the width. The area is 25 square meters. Let w be the width. So, the length of the rectangle is. Use the area formula for a the rectangle. Therefore, the length is or 3.5. Use the formula for perimeter of a rectangle. The perimeter of the rectangle is about 21.2 m. esolutions Manual - Powered by Cognero Page 3
4 36. The area of a circle is 25π square units. Use the area formula for a circle with radius r. The radius of the circle is 5 units. Find the circumference. Use the formula for the circumference of a circle with radius r. The circumference of the circle is 10π or about 31.4 units. 37. The area of a circle is 32π square units. Use the formula for the area of a circle with radius r. The radius of the circle is about units. Find the circumference. Use the formula for the circumference of a circle with radius r. The circumference of the circle is or about 35.5 units. esolutions Manual - Powered by Cognero Page 4
5 38. A rectangle s length is 3 times its width. The area is 27 square inches. Let w be the width. So, the length of the rectangle is 3w. Use the formula for the area of a rectangle. Therefore, the length is 3(3) or 9 in. Substitute in the formula for perimeter. The perimeter of the rectangle is 24 in. esolutions Manual - Powered by Cognero Page 5
6 39. A rectangle s length is twice its width. The area is 48 square inches. Let w be the width. So, the length of the rectangle is 2w. Use the formula for the area of the rectangle. Therefore, the length is in. Substitute in the formula for perimeter. The perimeter of the rectangle is esolutions Manual - Powered by Cognero Page 6
7 CCSS PRECISION Find the perimeter and area of each figure in inches. Round to the nearest hundredth, if necessary. 40. Before finding the perimeter and area, you must find the lengths of the two missing sides of the right triangle. In the diagram, it is indicated that the base is congruent to the height, so b = 2.5 cm. Use the Pythagorean Theorem to find the length of the hypotenuse. The perimeter of the triangle is the sum of the sides. Use dimensional analysis to change centimeters to inches. The area of the triangle is half the product of the base and the height. Use dimensional analysis to change cm 2 to in 2. So, the perimeter is about 3.42 in. and the area is 0.5 in 2. esolutions Manual - Powered by Cognero Page 7
8 41. Use the formulas to find the perimeter and area of the square. Use dimensional analysis to change from yards to inches. Use dimensional analysis to change yd 2 to in 2. So, the perimeter is 108 in. and the area is 729 in 2. esolutions Manual - Powered by Cognero Page 8
9 42. Use the formulas to find the perimeter and area of the rectangle. Use dimensional analysis to change feet to inches. Use dimensional analysis to change ft 2 to in 2. So, the perimeter is in. and the area is in Multiple Representations Collect and measure the diameter and circumference of ten round objects using a millimeter measuring tape. a. Tabular Record the measures in a table as shown. b. Algebraic Compute the value to the nearest hundredth for each object and record the result. c. Graphical Make a scatter plot of the data with d-values on the horizontal axis and C-values on the vertical axis. d. Verbal Find an equation for a line of best fit for the data. What does this equation represent? What does the slope of the line represent? a-b. Sample answer: Make a copy of the table given for the problem. Find ten circular objects, then measure the esolutions Manual - Powered by Cognero Page 9
10 diameter and circumference of each object to the nearest tenth of a centimeter, and record the results in the table. Divide the circumference by the diameter for each object and record the result in the table. For example, an object has a diameter of 3 cm and a circumference of 9.4 cm.. c. Choose a scale for your horizontal and vertical axis that will contain all your diameters and circumferences. Plot the ten points determined by each pair of diameter and circumference measures. d. Sample answer: Enter your data into a graphing calculator. In the STAT menu, enter the diameters into L1 and the circumferences into L2. Then in the STAT menu choose CALC and the LinReg(ax + b) function to find the equation for the regression line. Here, a 3.14 and b 0, so an equation for a line of best fit would be C = 3.14d; the equation represents a formula for approximating the circumference of a circle. The slope represents an approximation for pi. esolutions Manual - Powered by Cognero Page 10
11 Determine whether each statement can be assumed from the figure. Explain. 54. KJN is a right angle. Yes; the symbol denotes that KJN is a right angle. 55. PLN NLM It might appear that the angles are congruent but there are no marks on the angles in the diagram to indicate this. There is also nothing in the diagram that would provide information to know that bisects PLM. Because we do not know anything about the measures of these two angles, we can not assume that PLN NLM. 56. PNL and MNL are complementary. No; we do not know whether 57. KLN and MLN are supplementary. Yes; they form a linear pair. MNP is a right angle. esolutions Manual - Powered by Cognero Page 11
12 58. TABLE TENNIS The diagram shows the angle of play for a table tennis player. If a right-handed player has a strong forehand, he should stand to the left of the center line of his opponent s angle of play. a. What geometric term describes the center line? b. If the angle of play shown in the diagram measures 43, what is m BAD? a. The center line divides the angle into two congruent angles. So, it is an angle bisector. b. Since the center line is an angle bisector, esolutions Manual - Powered by Cognero Page 12
13 59. Name an appropriate method to solve each system of equations. Then solve the system. The appropriate method to solve this system is the elimination method, since neither of the equations have variables with coefficients of 1 and equations can not be simplified. Multiply the first equation by 2 and the second equation by 5 Substitute the value of y in one of the given equations. The solution of the system is: x = 3, y = 1 esolutions Manual - Powered by Cognero Page 13
14 60. The appropriate method to solve this system is graphing since both equations are in slope-intercept form and can be easily graphed. Graph the lines on a coordinate grid. The graphs appear to intersect at (3, 8). So, the solution of the system of equations is x = 3, y = 8. esolutions Manual - Powered by Cognero Page 14
15 61. The appropriate method to solve this system is substitution, since equation 1 has a variable with a coefficient of 1. Solve the first equation for x. Substitute this in the second equation. Substitute the value of y in one of the given equation. The solution to the system is x = 4, y = 2.5 esolutions Manual - Powered by Cognero Page 15
16 Evaluate each expression if P = 10, B = 12, h = 6, r = 3, and necessary. = 5. Round to the nearest tenth, if 62. Substitute. 63. Substitute. 64. Substitute. 65. Substitute. esolutions Manual - Powered by Cognero Page 16
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