2015-2016 Honors Geometry B Review

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HONORS GEOMETRY BSemester Exam ReviewHonors Geometry B2015-2016Exam ReviewNotes to the student: This review prepares you for the semester B Honors Geometry Exam. Theexam will cover units 3, 4, and 5 of the Honors Geometry curriculum.The exam consists of two parts. Part 1 is 15 selected response items each worth 2 points. Thesecond part is 15 extended response items worth varying numbers of points for a total of 70points.A calculator may be used on both parts of the exam.Many answers will be in terms of or in radical form. You may leave answers in that form orapproximate as a decimal to two places past the decimal point. For most accurateapproximations use the key on your calculator. Do not use 3.14 for . On the next two pagesyou will find the formulas that will be available to you in the exam booklet.FIGURES ARE NOT DRAWN TO SCALE!!!!! MCPSPage 1

HONORS GEOMETRY BSemester Exam ReviewArea/Circumference1Triangle: A bh2Rectangle: A bhParallelogram: A bh1Regular Polygon: A apothem perimeter2Circle Area: A r 2Circle Circumference: C 2 r dTrapezoid: A 1 b1 b2 h2VolumePrism/Cylinder:V Bh area of base heightPyramid/Cone:11V Bh area of base height33Sphere:4V r33Density FormulaDensity Mass VolumeCoordinate GeometrySlope:y2 y1x2 x1 x x y y2 Midpoint: 1 2 , 1 2 2Distance: x2 x1 y2 y1 22Conic Sections – Circle and ParabolaEquation of a circle with center at h, k and radius r: x h y k 22 r2Equations of a parabola with vertex at h, k with p the distance from vertex to focus and vertexto directrix : x h 2 y k MCPS2 4 p y k or y k 12 x h ;4p 4 p x h or x h 12 y k ;4popens up if p 0 , opens down if p 0 .opens right if p 0 , opens left if p 0 .Page 2

HONORS GEOMETRY BSemester Exam ReviewConic Sections – Ellipse and HyperbolaEquations of an ellipse with center at h, k , a b :Major axis horizontal: x h a2 y k 2b2 x h Major axis vertical:22b2 y k 12a2 1Equations of a hyperbola with center at h, k :Transverse axis horizontal: x h a2 y k 2 1b2 y k Transverse axis vertical:22a2 x h b22 1CirclesArc Length (degrees): S x 2 r 360Arc Length (radians): S r Sector Area (degrees): A x r2 360Sector Area (radians): A r22 Angle and Arc FormulasBENALDHGCm A 1 radian MCPS180 1 mBC2degreesJKFMm GDE 1 mGE mFH21 degree 180 m J 1 mMN mLK2 radiansPage 3

HONORS GEOMETRY BSemester Exam ReviewUnit 31.Which pairs of figures have the same volume?a.b.c.5r 4555Bases are squaresCross sections equidistantfrom the bases arecongruent and the heightsof the solids are equal.For items 2 through 4, find the volume if each figure is revolved about the dashed line.2.3.32455.4.673Name the cross section that is formed in each case.a.A cone, cut by a plane parallel to the base, not through the apex (top).b.A cone, cut by a plane perpendicular to the base, through the apex (top).c.A rectangular pyramid, cut by a plane parallel to the base, not through the apex(top).d.A square pyramid, cut by a plane perpendicular to the base, not through the apex(top), through one side only.e.A cylinder, cut by a plane parallel to the base.f.A cylinder, cut by a plane perpendicular to the base. MCPSPage 4

HONORS GEOMETRY BSemester Exam Review6.What is the relationship between the volumes of a cone and a cylinder if the cone andcylinder have the same radii and heights?7.What is the relationship between the volumes of a pyramid and a prism, if the pyramidand prism have the same base areas and heights?8.A scoop of ice cream is in the shape of a sphere with radius 3 cm is placed in a cone thathas a radius of 2 cm and a height of 9 cm. If the ice cream melts into the cone, will theice cream overflow the cone? Show how you determined your answer.9.The scoop of ice cream in item 8 has a mass of 124 grams. What is the density (ing/cm3 ) of the ice cream?10.A movie theater sells popcorn in pyramid-shaped boxes whose base is a square of side 15centimeters and a height of 25 centimeters.a.What is the volume of the popcorn box?b.The density of popcorn is 0.03 grams per cubic centimeter. What is the mass ofthe popcorn in the box?c.When popcorn is popped, it is stored in a cube that is 45 centimeters on eachside. How many boxes of popcorn can be filled from this container? MCPSPage 5

HONORS GEOMETRY B11.Semester Exam ReviewThe three figures below have the same volume. Find the missing height in each case.a.b.h6123h233h 24h For items 12 through 14, find the volume of each solid described.12.A sphere with radius 6 cm.13.A cone with radius 5 cm and height 9 cm.14.A square pyramid with base of side 8 cm and height 3 cm. MCPSPage 6

HONORS GEOMETRY B15.Semester Exam ReviewA company is producing a special part for a machine. The part consists of a cylinder oftin (white) that is inside of another cylinder made of copper (shaded). The part is shownbelow.6 mmCopper5 mm4 mm3 mmTina.What is the total volume of the entire part? You may give your answer in termsof or to the nearest cubic millimeter.b.What is the volume of the tin used in the part? You may give your answer interms of or to the nearest cubic millimeter.c.What is the volume of copper used in the part? You may give your answer interms of or to the nearest cubic millimeter. MCPSPage 7

HONORS GEOMETRY BSemester Exam ReviewUnit 4Use the word bank to complete the following definitions. Words may be used more than once.Word stantfoci16.A circle is the set of points in the plane that are from a given point,called the . The distance from the given point to every point on thecircle is called the .17.A parabola is a set of points in the plane that are from a given point,called the , and a given line, called the .18.An ellipse is a set of points in the plane such that the of the distances fromeach point on the ellipse to two fixed points, called the , is the same.19.A hyperbola is a set of points in the plane such that the of the distances fromeach point on the hyperbola, to two fixed points, called the , is the same.For items 20 and 21, sketch graphs of the following circles.20.x2 y2 421. x 2 2yO MCPS y 1 92yxOxPage 8

HONORS GEOMETRY BSemester Exam ReviewFor 22 and 23, write equations for the following circles and state the circumference and area ofeach nce:Circumference:Area :Area:Write the equation of a circle whose center is at the point 1,5 with the point 7, 20 on the circle.25.22Is the point 5,3 on the circle whose equation is x 3 y 2 65 ? Show howyou determined your answer. MCPSPage 9

HONORS GEOMETRY BSemester Exam Review22For each equation of a circle, complete the square to put it in the form x h y k r 2 ,then give the center and the radius.26.x 2 y 2 6 y 2727.x 2 2 x y 2 16 y 35 0For items 28 through 30, sketch graphs of the following parabolas. For each graph, also graphthe focus and directrix.28.x2 6 y29.y 2 8xyO MCPS30. x 2 2 4 y 1 yxOyxOxPage 10

HONORS GEOMETRY BSemester Exam ReviewFor items 31 and 32, write the equation for the parabola.31.32.yyF 1.5, 2 F 0, 2 O33.OxxLook at the parabola below.y10987654F321‐10 ‐ 9 ‐ 8 ‐ 7 ‐ 6 ‐ 5 ‐ 4 ‐ 3 ‐ 2 ‐ 1O12345 6789 10x‐1‐2‐3‐4‐5‐6‐7‐8‐9‐10Show that the point P 8,8 satisfies the definition of the parabola by showing that thedistance between the focus and the point P 8,8 is equal to the distance from the pointP 8,8 to the directrix. MCPSPage 11

HONORS GEOMETRY BSemester Exam ReviewFor items 34 and 35, sketch graphs of the following ellipses.34.x2 y2 125 1635. y 1 92 x 3 4y2 1yOOxxFor items 36 and 37, write equations for the ellipses below.36.O38.37.yxyODetermine whether the point 2,1 is on the ellipse whose equation is MCPSxx2 y 2 1 .94Page 12

HONORS GEOMETRY BSemester Exam ReviewFor items 39 and 40, sketch the following hyperbolas.x2 y 2 141239.40.y2 x2 199yOyxOxFor items 41 and 42, write equations for the hyperbolas below.41.O MCPS42.yxyOxPage 13

HONORS GEOMETRY B43.Semester Exam ReviewHow many points of intersection do the graphs of the circle x 2 y 2 16 and theparabola y x 2 have? You may use the grid below to help you.yO44.x2How many points of intersection do the graphs of the circles x 2 y 2 4 and x 2 2 y 2 4have? You may use the grid below to help you.yO MCPSxPage 14

HONORS GEOMETRY BSemester Exam Review Look at the graph of line AB on the coordinate plane below.y10987B 10,8 654321‐10 ‐ 9 ‐ 8 ‐ 7 ‐ 6 ‐ 5 ‐ 4 ‐ 3 ‐ 2 ‐ 1O123456789 10x‐1‐2‐3‐4A 10, 7 ‐5‐6‐7‐8‐9‐1045.What is the length of AB ?46.Point C is between points A and B and has coordinates 2, 2 . What is the ratio AC : CB ?47.Determine the coordinates of point D so that the ratio AD : DB 1: 4 . MCPSPage 15

HONORS GEOMETRY B48.Semester Exam ReviewLook at line k whose equation is y 3x 2 on the coordinate plane below.yOkxWrite equations for the following lines.a.Parallel to line k, passing through the point 0, 2 .b.Perpendicular to line k, passing through 0, 5 . MCPSPage 16

HONORS GEOMETRY BSemester Exam ReviewItems 49 through 52 (this item continues on the next page) uses the quadrilateral below.yA 1, 4 B 5,1 D 2,0 OxC 2, 3 Kelly thinks that the figure might be a square. Remembering her geometry she does thefollowing.49.Kelly first needs to know if the figure is a parallelogram. Show that the figure is aparallelogram by showing that the slopes of opposite sides are equal.50.Kelly now wants to determine if the figure is a rectangle. There are two ways to do this.a.Show, by using slopes, that one of the angles is a right angle, and therefore thereare four right angles and the figure is a rectangle.b.Show that the diagonals of the parallelogram are congruent, and the figure is arectangle. MCPSPage 17

HONORS GEOMETRY B51.52.Semester Exam ReviewNow that Kelly knows the figure is a rectangle. If she can show that it is a rhombus thenit must be a square. Again there are two ways to do this.a.Show that all four sides are the same length, and therefore the figure is a rhombus.b.Show, by using slope computations that the diagonals are perpendicular, andtherefore the figure is a rhombus.Kelly also remembers a property of parallelograms that the diagonals bisect each other.Show that this true by determining the midpoint of each diagonal and showing that eachmidpoint has the same coordinates. MCPSPage 18

HONORS GEOMETRY BSemester Exam ReviewFor items 53 and 54, determine the area and perimeter of each shaded figure below. Each gridline represents 10 meters.53.54.yy80806050201020100 10 20 30 40 50 60 70 80 90 1000 10 20 30 40 50 60 70 80 90 100 xxArea:Area:Perimeter:Perimeter:Unit 5For items 55 through 63, find the value of x in each figure below.55.56.57. C is the centerox50o160ooxCxo70o MCPSPage 19

HONORS GEOMETRY B58.Semester Exam Review60. AB is a x20o40o70o35oxo64.Quadrilateral ABCD is inscribed in the circle below.ABDCa. m A m C . Explain your reasoning.b. If AC is a diameter, what is m D and m B ? Explain your reasoning. MCPSPage 20

HONORS GEOMETRY BSemester Exam ReviewLook at the circle O below.A140o20o60oOBFE42oCDetermine the following:D65.m AB66. mEF67. mBC68.m AFB69.m FDB70.m FEA MCPSPage 21

HONORS GEOMETRY BSemester Exam ReviewLook at circle P below. Line m is tangent to the circle at point R.PR 8, PS 10, UW TZTmXWSURPZComplete the following.71.m PRS 73.TU is congruent to which segment?74. WTU is congruent to what angle?75. is congruent to which arc?TW76.SX 72.SR 77.What part of the circumference of a circle is represented by arc whose measure is radians?78.What part of the circumference of a circle is represented by an arc whose measure is 3radians? MCPSPage 22

HONORS GEOMETRY BSemester Exam ReviewJerald is creating a new target game. The game involves concentric circles (circles with the samecenter P) and sectors as shown below.The radius of the smaller circle is 6 inches, while the radius of the larger circle is 9 inches. is 12 inches.The length of DEThe measure of APC is 40o. The measure of JPH is 90o.BDPJEAHC79.AB ?What is the length of 80.What is the radian measure of APB ?81.What is the area of sector EPD ?82.What is the length of AC ?83.What is the area of sector EPH ?84. ?What is the measure of the circle segment bounded by JH and JH MCPSPage 23

HONORS GEOMETRY B85.Semester Exam ReviewA piece of jewelry is a segment of a circle with radius 10 mm, as shown in the shadedregion below.90o10 mm10 mmWhat is the area, in square mm, of the shaded region? Your answer may be in terms of or rounded to two decimal places. If you use in a calculation, use the calculator’s key. MCPSPage 24

Notes to the student: This review prepares you for the semester B Honors Geometry Exam. The exam will cover units 3, 4, and 5 of the Honors Geometry curriculum.

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