Geometry in Construction. 2. Graph the equations below. a) b) y = 3x + 1 y = -2x + 3

Transcription

1 Unit 4 Geometry in Construction Name Block 1. The LHS River Watch club wants to fence in an area along the Thompson riverbank to protect some endangered riverbank. The club has enough money to buy 200 feet of fencing to enclose a rectangular space. What is the area of the largest rectangle that can be enclosed? 2. Graph the equations below. a) b) y = 3x + 1 y = -2x A person who smokes cigarettes loses an average of 8 minutes of life for every cigarette smoked (actual statistic). If a person smokes 20 cigarettes a day, how many days per year is the person s life shorten? 4. Read the statement A certain quadrilateral, called ABCD, is not a parallelogram. Which conclusions can be made from the statement? a) Quadrilateral ABCD is a rectangle. b) Quadrilateral ABCD is a trapezoid. c) Quadrilateral ABCD is not a rhombus. d) Quadrilateral ABCD is a square. Page 1

2 5. A power saw was displayed in a store with a price tag of $70. No one seemed interested in buying it. The store manager decided to reduce the price by 10%. Still no one purchased the saw. The manager thought, If the saw won t sell at 10% off, I might as well raise the price again. So, he raised the price by 10%. What was the cost of the saw after these two price adjustments? Explain. 6. In the diagram at the right, BD is 5 in. long, find EG. ABD AEG. AD is 9 while AG is Jessi fenced a rectangle and then divided it into four sections. She then determined the area of three of the sections, as shown below. What is the area of the fourth section? 8. If a room has an area of 144 square feet, would it be more economical to frame a room that is square or a room that is rectangular? Explain using a counter example. 9. Find the length (exact value) of the diagonal of the house shown to make it a rectangular. Page 2

3 10. This is the floor plan of a room. Find the most economical way to frame this room. Fill in the table with your calculations. Remember there are 4 walls. 11. Find the length of the diagonals in the rectangular foundations. a) b) 12. Simplify a) 2x+ 3x b) 2 2 3a 5a c) 6x + 5x 3 2 d) 6x + x e) x + 6x 9x f) 13x 10x 2x Page 3

4 13. A fascia board was cut as shown. What should the angle of the other cut (y) be if you want the ends parallel? 14. A plumber is routing pipe through a home as shown below. What angle must the plumber use to keep the pipe parallel? 15. Aaron uses a simple device called a turnbuckle to square up structures that are parallelograms. For the gate pictured at the right, Aaron can tighten or loosen the turnbuckle on the diagonal cable. Why does this work.explain using geometry. 16. A quadrilateral is formed by the beams of the bridge at the left. a) What type of quadrilateral is it? b) Find the measures of the other interior angles of the quadrilateral. Page 4

5 17. A part of the stair rail forms a parallelogram ABCD because the spindles are parallel and the top and bottom rails are parallel. a) Name two pairs of congruent sides and two pairs of congruent angles in the parallelogram. b) Explain how you would know that the spindles are parallel. 18. The most common type of design for a house roof is a gable roof. The illustration shows the structural elements of a gable roof. Which structural element is a perpendicular bisector? Page 5

6 19. One type of bridge, a cable-stayed bridge is shown. a) Suppose m ABC = 110, what is m BAH? b) Suppose m DEG = 44, what is m DEF? c) Give the geometric name for the cable stay anchorage. d) From the construction side, why is DEG GEF 20. Find the missing angles. 21. A hip roof slopes at the ends of the building as well as the front and back. The front of this hip roof is in the shape of an isosceles trapezoid. If one angle measures 30, find the measures of the other three angles. Page 6

7 22. Find the missing angle measures x, y, and z in each trapezoid. a) b) 23. Whole-house fans are designed based on the interior square footage of living space in a home. Find the total area of each room below. 24. What is the degree measure of each angle between the 5 blades of a ceiling fan? 25. Multiply: a) 3(2 x x 9) b) ( x+ 5)(3x 8) c) (6 x)(3+ 2 x) d) (3x+ 1)(3x 1) e) ( x+ 5)( x 5) f) ( x+ 5)( x+ 5) 26. Carpenters use parallel wall studs in building supports for walls. Describe three ways a carpenter could guarantee that the wall studs AB and CD are parallel. Page 7

8 27. This logo is to be enlarged by a scale factor of 6. a) If the tree was originally 1 inch tall, what will the height of the new tree be? b) Will the original and the new logo be congruent or similar? c) If it takes 2 ml of color ink to print the small tree, how much will it take to print the enlargement? 28. Builders use properties of diagonals to square up rectangular shapes. Explain how to use properties of diagonals to locate the four corners. 29. Find the missing values. a) b) 30. Describe how you would square the sub-floor on this foundation. Page 8

9 31. Connor and Alex are each renting a trencher from 2 different rental companies. Connor is paying a $80 delivery fee and $50 per hour while Alex is paying no delivery fee but $65 per hour. a) Write an equation to represent what Connor is paying. b) Write an equation to represent what Alex is paying. c) Graph both equations on the graph below. d) At what point do the two lines intersect? e) What does this point represent? 32. An isosceles trapezoidal window is ordered from the factory. One angle is marked. What are the measures of the other 3 angles? 33. You have done a calculation and the calculator gave you 3.2 feet. How many inches is that? Page 9

10 34. The isosceles truss shown below has a vertex angle of 124 o. What is the measure of x? 35. Each side of a house plan is doubled. What happens to the perimeter? What happens to the area? 36. Below is a plot plan of a subdivision showing the roads. Find the angles created by the roads. a= b= c= d= e= f= g= h= i= 37. Sara is building a triangular pen for the pets. If two of the sides measure 7 feet and 10 feet, what is the range of lengths for the third side? Page 10

11 38. Trevor uses a circular saw to cut a piece of lumber at a 45 angle. What is the measure of the other angles formed by the cut? 39. Simplify and/or multiply a) 3(2x 1)( x+ 5) b) 2 (2x 3) c) 2 ( x + 7) d) (4 x) Find the length of the median (the middle support) of the trusses shown below. a) b) c) 41. Find the lengths of the bases and the median. a) b) Page 11

12 42. Larry has a ladder that is 16 long. If he sets the base of the ladder on level ground 5 feet from the side of the house, how many feet above the ground will the top of the ladder reach when it rests against the house? 43. The Washington Monument is an obelisk, a large stone pillar that gradually tapers as it rises, ending with a pyramid on top (not shown in drawing). Each face of the monument under the pyramid is a trapezoid. The monument s base is about 55 feet wide, and the width at the top, just below the pyramid, is about 34 feet. How wide is the monument at its median? 44. Find the value of x a) b) Page 12

13 45. Solve: a) 3( x 8) = 4( x+ 2) b) 9( x + 6) = Multiply a) 3(2x 1)( x+ 5) b) 2 2( x + 1) 47. Graph these coordinates: A(1,2),B (3,1),C (0,0),D (2,-1). Using the concept of slope, prove the opposite sides are parallel and adjacent sides are perpendicular. 48. Graph these coordinates: A(1,2), B(3,1), C(0, 0), D(2,-1). Using the distance formula, prove the sides are equal length. Page 13

14 49. Graph these coordinates: A(-2,1), B(-1,4), C(-1,0), D(0,3). Prove the figure is a rectangle using slope and distance formula. 50. Graph these coordinates: A(0,0), B(-2,4), C(2, 1), D(0,5), Is the figure a rectangle, square, or rhombus? Prove using slope and distance formula. 51. Graph these coordinates: A(1, -1), B(-2, -3), C(-4, 0), D(?,?). What is the fourth coordinate so that the figure is a square? Prove the figure is a square using slope and distance formula. Page 14

15 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 52. A segment congruent to the segment ABshown above. 53. A segment congruent to the segment AC shown above. 54. A segment congruent to the segment BC shown above. Page 15

16 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 55. Construct an equilateral triangle with a sides the length of AB 56. What size angle did you construct in the above problem? 57. Construct a perpendicular bisector to AB. Page 16

17 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 58. Construct a perpendicular to ST through Q. 59. Construct a perpendicular to AB thru point C Page 17

18 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 60. Construct any rectangle with side AB 61. Construct a square with side AB Page 18

19 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 62. Construct the midpoint of AB. 63. Construct the midpoint of ST. Page 19

20 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 64. Construct an angle congruent to ABC on the line provided. 65. Construct an angle congruent to XYZ on the line provided. Page 20

21 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 66. Construct a line parallel to line m passing thru point P. 67. Construct a line parallel to line m passing thru point P. Page 21

22 For # Using only a compass and a straightedge (not necessarily a ruler), construct: 68. Construct the angle bisector of XYZ. 69. Construct the angle bisector of ABC. Page 22

23 70. In a machine shop, you have been given the task of drilling 6 equally spaced holes around a central hub assembly (center). The sheet of metal is shown below. For #70, using only a compass and a straightedge (not necessarily a ruler), construct 6 equally spaced points on the circle: Page 23

24 71. Betsy is making a wooden frame to replace the one on an octagonal antique mirror. Determine the measure of each interior angle of the frame if its shape is a regular octagon. 72. The gable roof is shown. The roof and the walls of the building are models of planes. a) Name a point that is coplanar with points C and D. b) Name point that is noncoplanar with points R and S. c) Name one point that is in two different planes. 73. Find the value of x in each of these trusses. a) b) c) 74. In triangle CDE the measure of D is 37. Find the measure of each numbered angle. Page 24

25 75. The Cherokee people had a system of government based on seven clans. Their council house consisted of seven sides, one for each clan, and its shape was a regular heptagon. Find the measure of each interior angle of this figure. 76. The floor of a bathroom is covered using regular hexagonal tile. a) Find the area of one tile if each side is 4 inches and the apothem is 3.4 inches. b) What is the measure of ABC? 77. The people of the Artic use blocks of snow/ice to build igloos (hemisphere). Historically, these were their homes. Suppose you have a 15 diameter igloo. a) Find the floor area in the igloo to the nearest square foot. b) Find the surface area of the igloo to the nearest square foot (ignore the floor). c) Find the volume of the igloo to the nearest cubic foot. d) Find the floor area of a square home where each sides measures 13.3 feet. e) Find the surface area of the square home. Assume a flat roof, walls are 15 tall, and ignore the floor. Page 25

26 77. Continued f) Find the volume of the square home. g) With both homes having the same square footage, which one is easier to heat and why? 78. a) Is a rectangular house considered a square house? If false, give counter example. b) Is a square house considered a rectangular house? If false, give counter example. 79. Juanita must run a electric line between poles at opposite ends of the lake, as shown in the drawing. The length and width of the lake are 80 feet and 50 feet. What is the distance between the two poles, to the nearest foot? 80. The diameter of a dome house is tripled. a. What happens to the volume? b. If the original dome had a volume of 180 cu. ft, what is the volume of the larger dome home? Page 26

27 81. Last years cabin is similar to the drawing shown below. The rise (A) was 4.5 feet. B was 7.5 feet. a) What was the pitch of the roof (expressed as some number over 12)? b) What is the truss length marked c? c) The length of the cabin was 40. What is the area of the total roof? 82. The floor of the tipi is an 18 sided regular polygon. The apothem is 8ft and each side is 3 ft. a) What is the floor square footage? b) What is the volume of the tipi if the height is 20? 83. The roof below is made of 4 equal triangles. The area of the roof totals 2000 sq feet. The slant height of the roof is 10. What is the dimension of the square home? Page 27

28 84. Find the missing angles that the roads form in the map below. 85. You have heard the saying I have a million things to do. If you took 5 minutes to do one thing, how many days would it take you to do a million things? 86. On a coordinate graph, point (-6, -2) is translated (slid) to (2,3). If a second point is (-2,-4) is translated (slid) using the same transformation (movement), what is the location of (-2,-4)? 87. Hannah designed a game of chance. She played the game 40 different times the fist day and won 15 of them, She played the game 65 times the second day, winning 38. The last day, she played the game 28 times, and won 14. Using this information, what would you expect the probability of her winning this game. Express your answer as a fraction and a percent. Page 28

29 88. Logan spun the arrow on the spinner below 50 times and recorded the outcomes as follows: a) What is the experimental probability of getting a C? b) What is the theoretical probability of getting a C? 89. Mindy earned the following test scores in Geometry in Construction. Scores: Which of the following measures of central tendencies will be affected if Mindy earns a score of 65 on her next test? Circle one Median only Mean, median, and range Mean & median only Mean, median, mode, and range 90. This gable which is an isosceles triangle has one base angle of 40. What is the measure of the vertex angle? Page 29

30 91. Paul has designed a triangular deck shown below. He wants to place benches in the two larger corners. Which corners have the larger angles? 92. A kitchen work triangle is shown below. The vertices of the triangle are anchored by the cooking area, the cleaning area, and the refrigerator. What are the possible distances between the refrigerator and the range (find x). 93. Jason and Joey are building a pyramid using blocks that measure 6 by 6 by 6. The base is drawn below using 24 blocks. Each successive level contains blocks that overlap by 3 as partially shown in the second figure. If the pattern continues, how many total blocks will be needed to complete the pyramid. Page 30

31 94. If the sides of a shipping crate are made one-tenth the original length, what fraction of the original surface area is the new surface area? 95. areaofsmall 1 = Find x if the two figures are similar areaofl arg e DOG CAT The area of triangle DOG is 625 sq ft. Find the area of triangle CAT 97. Two similar hexagonal prisms have areas of 25 sq ft and 49 sq ft. What is the ratio of the corresponding side lengths? Of the perimeters? Page 31

32 98. If the sides of a model balsa wood house are made one-sixth of the original length, what fraction of the original area is the new area? 99. Two similar water tanks are shown below. a) Find the volume of the small tank b) Find the ratio of their volumes c) Find the ratio of their radii d) Find H 100. A storage garage shaped like a rectangular prism measures 8 ft by 10 ft by 12 ft. It rents for $150 per month. You have collected lots of stuff and will be renting one that has 8 times the volume and is similar to the first storage garage. Find the dimensions of your new storage garage Scotty is building a statue that is 4 inches tall. It weighs 5.1 lbs. She has been hired to build a similar statue that will be 4 ft tall. What will it weigh? 102. Shay has a small barn that has a volume of 600 cu ft. She is going to build a new barn similar to the old. The new barn is going to have 27 times the volume. How will the dimensions compare? Page 32

33 103. The volumes of two similar silos are 729 cubic feet and 512 cubic ft. The surface are of the larger silo is 160 sq ft. What is the surface area of the smaller silo? 104. If a warehouse that measure 30 ft by 60 ft by 8 ft requires a 6 ton air conditioner, what size of air conditioner unit will a similar building 3 times the dimension require? 105. The building in #104 above requires 10 gallons of paint to paint the exterior. How many gallons of paint will the similar building 3 times the dimension require? 106. The building in #104 above requires $500 in baseboard molding. How much will you spend on the new similar building 3 times the dimension? 107. Your calculator gives the answer of 5.8 feet. What is the measurement you would use on the tape measure? Page 33