MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 3 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

Vocabulary: chord, central angle, inscribed angle, arc, minor arc, major arc, semicircle, adjacent arcs, Inscribed Angle Theorem, Inscribed Quadrilateral Theorem, tangent, point of tangency, Chord-Chord Product Theorem, secant, secant segment, external secant segment, Secant-Secant Theorem, tangent segment, Intersecting Chords Angle Measure Theorem, Tangent-Secant Interior Angle Measure Theorem, Tangent-Secant Exterior Angle Measure Theorem

Geometry – Intensive Geometry

REPORTING CATEGORY PRACTICE ITEMS 1. James’ math class compares the diameters and circumferences of multiple circular objects. They create a table of values and a graphical representation of the data in order to

investigate the relationship between the two measurements. What will the class notice about the representations? How does this relate to what you know about circles? 2. The medieval labyrinth shown below consists of concentric circles. Justify that any of the circles are similar to each other.

Source: https://www.ncetm.org.uk/resources/25811 3. Complete the proof below.

Given: Circle C with radius r and circle D with radius s. Prove: Circle C is similar to circle D. a. First transform circle C with the translation that maps point C onto _________. b. Let the image of circle C be circle C′. The center of circle C′ must lie at _________. c. Now transform circle C′ with a dilation that has center D and scale factor _________. d. After the dilation, the radius of the image of circle C′ is _________. e. The center of the image of circle C′ is _________.

Since translations and dilations are _____________________________, you can conclude that circle C is similar to circle D.

Reporting Category: Circles, Geometric Measurement, and Geometric Properties with Equations % of Test Average % Correct 2015 Average % Correct 2016

38% 36% 19%

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 4 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 4. Prove the circles made by the bike’s wheels and the gear are similar by taking measurements, then using transformations and/or algebraically.

5. Circle A with radius 4 and center (3, 0) is drawn in the coordinate plane. What is the scale factor that maps the circle with radius 3 and center (2, 3) onto circle A?

6. A circular waterfall is surrounded by a brick wall. The radius of the inner wall is 35 inches. If the bricks are 5 inches wide, what scale factor was used to determine the radius of the outer wall?

7. Prove that circle A with center (2, 1) and radius 4 is similar to circle B with center (-1, -1) and radius 2.

8. Refer to diagram shown of Circle A (assume that lines that appear tangent are tangent)

a) Name 3 chords b) Name any diameters c) Name a circumscribed angle d) Name a central angle e) Name a major arc f) Name a minor arc g) Name a semi-circle h) Name two angles whose measures

are in a ratio of 1:2 i) Name 3 vertices that would form a

right triangle j) Name 3 vertices that would a different

right triangle

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 5 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 9. Refer to diagram above of Circle A (Assume that lines that appear tangent are tangent). Determine if Column I is <, > or = to Column II

Column I <, > or = Column II Justify EA GA Measure of arc EB Measure of arc FC Measure of ∠EAF Measure of ∠EDF

10. Refer to diagram above of Circle A (Assume that lines that appear tangent are tangent). If the measure of ∠FAC = 50°, find the measures of the following: a) Arc BC b) Angle BHC c) Angle EDC d) Arc CEB

11. Write the equation for a line tangent to circle A. Explain how you determined your equation.

12. In the accompanying figure of circle O, chords 𝐴𝐴𝐴𝐴���� and 𝐶𝐶𝐶𝐶���� intersect at E and 𝐴𝐴𝐶𝐶���� is a diameter. If 𝑚𝑚𝐶𝐶𝐴𝐴� = 82, find 𝑚𝑚∠𝐴𝐴𝐴𝐴𝐶𝐶.

13. The accompanying diagram shows a child’s spin toy that is constructed from two chords intersecting in a circle. The curved edge of the larger shaded section is one-quarter of the circumference of the circle, and the curved edge of the smaller shaded section is one-fifth of the circumference of the circle. What is the measure of angle 𝑥𝑥?

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 6 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 14. In the circle given, 𝐴𝐴𝐶𝐶���� is the diameter and is perpendicular to chord 𝐶𝐶𝐴𝐴����. 𝐶𝐶𝐷𝐷 = 8 cm, and 𝐷𝐷𝐴𝐴 = 2 cm. Find 𝐴𝐴𝐶𝐶, 𝐴𝐴𝐶𝐶, 𝑚𝑚∠𝐶𝐶𝐴𝐴𝐴𝐴, the arc length of 𝐶𝐶𝐴𝐴𝐴𝐴�,

and the area of sector 𝐶𝐶𝐴𝐴𝐴𝐴 (round to the nearest hundredth, if necessary).

15. Given circle A with ∠BAC≅∠BAD, find the following (round to the nearest hundredth, if necessary) a) 𝑚𝑚𝐶𝐶𝐶𝐶� b) 𝑚𝑚𝐶𝐶𝐴𝐴𝐶𝐶� c) 𝑚𝑚𝐴𝐴𝐶𝐶𝐶𝐶� d) Arc length 𝐶𝐶𝐶𝐶� e) Arc length 𝐶𝐶𝐴𝐴𝐶𝐶� f) Arc length 𝐴𝐴𝐶𝐶𝐶𝐶� g) Area of sector 𝐶𝐶𝐴𝐴𝐶𝐶

16. Given circle A with 𝑚𝑚∠𝐶𝐶𝐴𝐴𝐶𝐶 = 50°, a) Name a central angle. b) Name an inscribed angle. c) Name a chord. d) Name a minor arc. e) Name a major arc. f) Find 𝑚𝑚𝐶𝐶𝐶𝐶� g) Find 𝑚𝑚𝐶𝐶𝐴𝐴𝐶𝐶� h) Find 𝑚𝑚∠𝐶𝐶𝐴𝐴𝐶𝐶.

17. Find 𝑚𝑚𝐴𝐴𝐴𝐴� and 𝑚𝑚𝐶𝐶𝐴𝐴�

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 7 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 18. A space station orbiting the Earth is sending two signals that are tangent to the Earth. If the intercepted arc of the Earth’s surface that is visible to the satellite is 168º, what is the

measure of the angled formed by the two signal beams? 19. You are using binoculars to look at an eagle. The sides of the binoculars seem to extend from the eagle, and are tangent to the circular portion. How far away is the eagle?

20. In the map of Georgia on the right, Interstates 16 and 75 and Route 280 form a triangle with Macon as one of the vertices. A company wants to build its new headquarters in the middle of that triangle so that the building is equidistant from each highway. Where should the headquarters be built?

21. Jane, Keith, and Lee are shopping at a mall. The halls are arranged in a star shape, as shown in the diagram. The friends’ locations are marked on the diagram. Through texting, they arrange to meet up so they can grab a cinnamon bun. Where should they meet so that each person has the same distance to walk?

22. In planning a new technology building for a college, an architect needs to make sure that the server for the computer network will be in a room that is equidistant from three computer labs. In which room should the server be placed?

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 8 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 23. A cell phone company wants to install a tower between Routes 108, 216, and Hall Shop Road that will boost the existing signal.

a) Construct the approximate point at which the company should place a cell phone tower. Explain your strategy, including any assumptions you may have made. b) When the company tests the tower, its strength does not reach beyond the indicated roads. Your house is 1,000 feet from the tower. Will you receive the additional boost to

your signal?

24. Which of the following best describes how to find the circumcenter of a triangle?

A. Find the intersection of the three altitudes of the triangle. B. Find the intersection of the perpendicular bisectors of the sides of the triangle. C. Find the intersection of the angle bisectors for each angle in the triangle. D. Find the intersection of the three medians of the triangle.

25. Use the diagram to explain why m∠ADC = 85°.

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 9 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 26. If a standard dartboard’s diameter is 17.75 inches, what is the area of one sector?

Image Source: https://upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Dartboard_unlabeled.svg/453px-Dartboard_unlabeled.svg.png 27. Using the following maps and information, answer the questions below:

Diameter of Circle: 385 feet Source: Google Maps

If Washington, DC wishes to repave Logan Circle (pictured above) the contractors will need precise measurements. a) What is the total distance around the circle? b) If contractors need replacing the inner curb (no break) from one intersection of Rhode Island to the other, how much would that be? c) What length of curb would be needed between the sidewalks at P St NW (A) and 13th St NW (B)? d) If the contracting company causes problems with the grass in the circle between the sidewalks at 13th St NW (B) and Rhode Island Ave NW (C), how many square feet of

landscaping would they have to pay for? 28. The pendulum of a clock swings through an angle of 2.5 radians as its tip travels through an arc of 50 centimeters. Find the length of the pendulum, in centimeters. 29. In a circle, a central angle of 2 radians intercepts an arc of 6 centimeters. Find the length of the radius in centimeters. 30. A central angle of a circular garden measures 2.5 radians and intercepts an arc of 20 feet. What is the radius of the garden?

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 10 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 31. Solve for x. Assume that lines which appear tangent are tangent.

32. The picture below shows a field that has been treated with a pivot irrigation watering system and an example of a pivot irrigation system. A long arm extends from the center (Point

O) to the edge of the field.

Word Bank

a) Given that the diameter is 1 mile, OD, which is the ____________, is 0.5 miles. To measure the distance around the irrigated area, or the ____________, a farmer would use

the formula ____________, which will be approximately ___________ miles.

b) Since m∠AOC = ___________, the distance around the circumference from A to C, known as the ___________, will be ___________ miles, or approximately _________ feet. c) The area of the sector DOC can be found by taking ___________ divided by 360°, then multiplied by the ___________, which will be approximately ___________ square miles. d) Calculate the area of sector DOC in square feet.

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Radius

Circumference

3.14

Area 120º

0.044 Center

5528.16

Sector Area

Arc Length

𝐶𝐶 = 2𝜋𝜋𝜋𝜋

20º

Central Angle

1.047

MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 11 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 33. Complete the square to find the center and radius of a circle given by the equation: 𝑥𝑥2 + 6𝑥𝑥 + 𝑦𝑦2 − 14𝑦𝑦 + 42 = 0 34. Find the equation of circle A.

Find the points of intersection of the circle and the x-axis (round to the nearest hundredth)

35. Write the equation for the circle shown below:

36. A landscape company has been contracted to build a circular garden with diameter 20 feet. They plan to place a fountain at

its center. The yard has been graphed with vertices (0,0), (19,0), (19,11), and (0, 11). The house’s coordinates are (3,1), (3, 7), (13, 1), and (13, 7). One unit on the graph equals five feet. Write an equation for the border of the garden, and state the coordinates for the fountain. The garden’s border may not touch the house.

37. A farmer tethers her goat to a stake with a rope 15 feet long. Every day she must move the stake so the goat can eat fresh

grass. She has a small herb garden whose vertices are (2, 1), (6, 1), (6, 6), and (2, 6). If she moves the stake to the points listed below, state whether the goat can nibble on the herb garden. Stake coordinates:

I. (20, 12) II. (18, 12)

III. (16, 1)

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 12 of 21 Topic VII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 38. Are the points (4, 4), (0, -3), and (5,2) on, inside, or outside the circle whose diameter is defined by (-3,2) and (5, -6)? 39. The following shows a floor plan for a gazebo.

Source: http://www.lancastercountybackyard.com/floor-plans/dodecagon-gazebo-floor-plan

a) Describe how a circle could be used to estimate both the area and perimeter of the gazebo. b) Describe a method that would not use a circle to estimate the area and perimeter. c) Assume the image is drawn to scale and that 1 cm = 1 foot. Use both of your methods above to find area and perimeter. d) How do they compare? Which do you think is most accurate? Which method was easier? Is there a time when you could imagine using each?

40. Using a paper plate, demonstrate how “slices” of a circle can be rearranged to approximate a parallelogram. How might the formula for the area of the circle be derived from this approximation?

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 3 of 18 Topic VIII Third Nine Weeks

GEOMETRY Course Code: 120631001

MODELING CYCLE (★) The basic modeling cycle involves:

1. Identifying variables in the situation and selecting those that represent essential features. 2. Formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the

variables. 3. Analyzing and performing operations on these relationships to draw conclusions. 4. Interpreting the results of the mathematics in terms of the original situation. 5. Validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable. 6. Reporting on the conclusions and the reasoning behind them.

Choices, assumptions, and approximations are present throughout this cycle. http://www.cpalms.org/Standards/mafs_modeling_standards.aspx

Vocabulary: right prism, right cylinder, oblique prism, oblique cylinder, cross section, pyramid, cone, sphere, net, surface area, slant height, density, scale factor. Geometry – Intensive Geometry

REPORTING CATEGORY PRACTICE ITEMS 1. Blocks of cheese can come in a variety of shapes and sizes. The form depends on the type of cheese and the process used to make it. Examine the various blocks of cheese

shown below. Name the three-dimensional object that best describes the block of cheese. List the dimensions of each block of cheese that you would need to determine the volume.

a)

https://antiochcriticalskills.files.wordpress.com/2013/04/cheese_oh_cheese.jpg

b)

http://www.ingersolltimes.com/2013/03/28/route-to-the-past-the-story-of-the-big-cheese c)

http://www.renardscheese.com/provolone/

d)

http://www.foodnutritiontable.com/nutritions/nutrient/?id=265

Reporting Category: Circles, Geometric Measurement, and Geometric Properties with Equations % of Test Average % Correct 2015 Average % Correct 2016

38% 36% 19%

Reporting Category: Modeling with Geometry % of Test Average % Correct 2015 Average % Correct 2016 16% 18% 25%

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 4 of 18 Topic VIII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS

2. A carpenter made a storage container in the shape of a rectangular prism. It is 5 feet high and has a volume of 720 cubic feet. He wants to make a second container with the same height and volume as the first one, but in the shape of a triangular prism. What will be the number of square feet in the base of the new container?

3. As shown in the diagram below, a landscaper uses a cylindrical lawn roller on a lawn. The roller has a radius of 9 inches and a width of 42 inches

To the nearest square inch, the area the roller covers in one complete rotation is A. 2,374 B. 2,375 C. 10,682 D. 10,688

4. In the diagram below, a right circular cone with a radius of 3 inches has a slant height of 5 inches, and a right cylinder with a radius of 4 inches has a height of 6 inches.

Determine and state the number of full cones of water needed to completely fill the cylinder with water.

5. If the surface area of a sphere is represented by 144π, what is the volume in terms of π? A. 36 π B. 48 π C. 216 π D. 288 π

6. The Zeiss Major Planetarium in Berlin, pictured at right, is one of the largest planetariums in the world. The dome of the main hall has a diameter of 23 meters. The dome is not a full sphere, as the bottom is truncated to maximize seating. If the volume of the planetarium dome is 5478.74 m³, what percent of a full sphere is the dome?

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 5 of 18 Topic VIII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 7. A wooden cube has an edge length of 6 centimeters and a mass of 137.8 grams. Determine the density of the cube, to the nearest thousandth. State which type of wood the cube

is made of, using the density table below.

8. A silicon wafer is a circular disc 80 millimeters in diameter. One side of the wafer is coated with 0.06 milligrams of a substance, called photoresist, to a uniform thickness.

Photoresist has a density of 1.2 milligrams per cubic millimeter.

a) What is the volume of photoresist used on the wafer? b) What is the thickness of photoresist on the wafer?

9. A construction company is preparing 10 acres of land for a new housing community. The land contains large rocks that need to be removed. A machine removes 10 rocks from 360

square feet of land. 1 acre = 43,560 square feet About how many rocks will need to be removed from the 10 acres of land?

10. Walter wants to make 100 candles in the shape of a cone for his new candle business. The mold shown below will be used to make the candles. Each mold will have a height of 8 inches and a diameter of 3 inches. To the nearest cubic inch, what will be the total volume of 100 candles?

Walter goes to a hobby store to buy the wax for his candles. The wax costs $0.10 per ounce. If the weight of the wax is 0.52 ounce per cubic inch, how much will it cost Walter to buy the wax for 100 candles? If Walter spent a total of $37.83 for the molds and charges $1.95 for each candle, what is Walter's profit after selling 100 candles?

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 6 of 18 Topic VIII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 11. Stephanie has an aquarium that is in the shape of a right rectangular prism, 50 centimeters long, 25 centimeters wide and 30 centimeters tall.

For decoration, Stephanie wants a layer of marbles in the bottom of the tank about 5 centimeters deep. The marbles have a diameter of 1 centimeter and come in bags of 500. a) How many bags of marbles will Stephanie need? b) Stephanie pours all of the marbles into the tank. She now adds water until its level is 3 centimeters below the top of the tank.

How much water is in the tank? Express your answer in liters (1 liter = 1000 cubic centimeters). 12. The map below shows the counties in the State of Nevada. The shaded area is Esmeralda County.

This diagram below shows Esmeralda County superimposed on a grid.

a) Approximate the area of Esmeralda County. b) Esmeralda County is the least populated county in Nevada with only 775 people. What is the population density of Esmeralda County?

13. The volume of a cylinder is 102 cm3. Use what you know about the formula for the volume of cylinders and cones to calculate the volume of a cone that has the same height and the same base radius as the cylinder. Explain how you determined your answer

14. How is determining the volume of a cone like determining the volume of a pyramid? How is it different? 15. The volume of a pyramid is 61 cm3. Use what you know about the formula for the volume of pyramids and prisms to calculate the volume of a prism that has the same height and

the same base as the prism. Explain how you determined your answer. 16. Five cubes of ice with a base edge measuring 3 cm are melting in a cylindrical glass that has a radius of 7 cm. How high will the liquid in the glass rise when all of the cubes have

melted? (Ignore the fact that ice takes up slightly more space than water). 17. Layla is working for a soft serve ice cream shop. The shop serves two different types of cones: a cake cone and a sugar cone. She wants to

know which type holds more ice cream in the cone. Both cones have the same height and a diameter of 3 inches at the opening of the cone. The cake cone consists of two cylinders, where the top cylinder is a third of the height of the entire cone. Use what you know about the volume formulas to make an argument as to which cone will hold the most ice cream.

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 7 of 18 Topic VIII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 18. Ms. Greene, the manager of Paradise Lake, just ordered a fish food dispenser like the one shown. She plans to use a paper cone to hold the pellets. The

machine is set to dispense no more than 118 cm3 of food for each token. The cones Ms. Greene plans to use are 8 cm in diameter. How tall must the cone be to hold the food?

https://www.gumball-machine.com/vending-machines/pet-food-vending-machines/coin-erated-fish-food-dispenser.html 19. Wally’s Winter Wonderland is offering a special workshop this holiday season. Patrons can design their own snowman snow globe. The snow globes come in

one standard size, a glass sphere with a 12 cm diameter atop a wooden base. Participants will be able to design the snowman that will be placed inside. The snowman’s body is made up of 3 spheres. In order for the glitter to properly circulate throughout the snow globe, at least 50% of the volume of the globe must be water. Design a snowman that can be placed in the standard snow globe.

20. You are the restaurant manager of Renaissance Times. You recently ordered the three new types of glasses for your guests, but

you cannot remember which glass you had decided would be a small, medium, or large. You do know all of the glasses have the same height, since they need to fit in the kitchen dishwasher. Use the dimensions of each glass below to determine which glass should be a small, which glass should be a medium, and which glass should be a large.

21. Bob takes his aluminum cans to the recycling center. The cylindrical cans are first dropped into a machine that slices them. Then they move down a conveyer belt into a machine

that crushes them. Bob notices that the cans can come out of the slicer in various forms depending on whether they are sliced vertically, horizontally, and diagonally. Draw a sketch of each of the cross-sections that could result depending on the angle at which the can is sliced

22. A three-dimensional object has a parallel cross-section that is a circle and a perpendicular cross-section that is an isosceles triangle. What could the three-dimensional object be? Name and/or draw some real-world objects that would have these qualities.

23. Figure 2 below was formed from figure 1 by keeping the length of segment BC unchanged and decreasing the measure of angle ABC.

In each figure, points B and C are fixed and segment AB is rotated in space around segment BC. Compare the 3-dimensional shapes created by each rotation

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 8 of 18 Topic VIII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 24. A rectangle will be rotated 360º about a line that contains the point of intersection of its diagonals and is parallel to a side. What three-dimensional shape will be created as a result

of the rotation? A. a cube B. a rectangular prism C. a cylinder D. a sphere

25. Paul is designing a mold for a concrete block to be used in a custom landscaping project. The block is shown in the diagram with its corresponding dimensions and consists of two intersecting rectangular prisms. Find the volume of mixed concrete, in cubic feet, needed to make Paul’s custom block.

26. A triangular prism has an isosceles right triangular base with a hypotenuse of √32 and a prism height of 15. A square prism has a height of 15, and its volume is equal to that of the

triangular prism. What are the dimensions of the square base?

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MIAMI-DADE COUNTY PUBLIC SCHOOLS District Pacing Guide

Division of Academics - Department of Mathematics Page 9 of 18 Topic VIII Third Nine Weeks

GEOMETRY Course Code: 120631001

REPORTING CATEGORY PRACTICE ITEMS 27. Two jars of peanut butter by the same brand are sold in a grocery store. The first jar is twice the height of the second jar, but its diameter is one-half as much as the shorter jar.

The taller jar costs $1.49, and the shorter jar costs $2.95. Which jar is the better buy? 28. A cone with base area 𝐴𝐴 and height ℎ is sliced by planes parallel to its base into three pieces of equal height. Find the volume of each section. 29. The frustum of a pyramid is formed by cutting off the top part by a plane parallel to the base. The base of the pyramid and the cross-section where

the cut is made are called the bases of the frustum. The distance between the planes containing the bases is called the height of the frustum. Find the volume of a frustum if the bases are squares of edge lengths 2 and 3, and the height of the frustum is 4. (not to scale)

30. A bulk tank contains a heavy grade of oil that is to be emptied from a valve into smaller 5.2-quart containers via a funnel. To improve the efficiency of this transfer process, Jason

wants to know the greatest rate of oil flow that he can use so that the container and funnel do not overflow. The funnel consists of a cone that empties into a circular cylinder with the dimensions as shown in the diagram. Answer each question below to help Jason determine a solution to his problem. a) Find the volume of the funnel.

b) If 1 in3 is equivalent in volume to 4

231qt., what is the volume of the funnel in quarts?

c) If this particular grade of oil flows out of the funnel at a rate of 1.4 quarts per minute, how much time in minutes is needed to fill the 5.2-quart

container?

d) Will the tank valve be shut off exactly when the container is full? Explain.

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