georgia online formative assessment resource (gofar)...

G9-C-GEO-Q2-42

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Georgia Online Formative Assessment Resource (GOFAR)

Copyright © 2015by Georgia Department of Education. Items shall not be used in a third party system or displayed publicly. Page: (1 of 35 )

1. Circle R and circle S intersect at two points. The graph shows the centers of the circles, R and S, and oneof their points of intersection, P.

Part AWhat is the equation of circle R? Explain your answer.

Part BWhat is the equation of circle S? Explain your answer.

Part CWhat is the second point of intersection of the circles? Explain your answer.

Be sure to complete ALL parts of the task.Write your answer and show your work on the paper provided.

Do NOT type your answer in the text box below.

2. Figure WXYZ is graphed on the coordinate plane.



Part AName a transformation consisting of one rotation about the origin followed by two reflections across theaxes that carries figure WXYZ onto itself. Use coordinates to show this is true. Explain your answer.

Part BDetermine whether the opposite sides of figure WXYZ are congruent. Show your work.

Part CDetermine whether the opposite angles of figure WXYZ are congruent. Show your work.

Part DDetermine whether the diagonals of figure WXYZ are congruent. Show your work.

Part EBased on Parts A–D, what is the best classification of figure WXYZ? Explain your answer.



3. The equation represents a circle in general form.

Part AFind the equation of the circle in standard form. Show your work and explain each step.

Part BWhat are the center and radius of the circle? Explain your answer.



Part CLeah says that the point is on the circle. Show work to prove Leah’s statement using both forms ofthe equation.

Part DPoints P and Qform diameter of the circle. Point Qis located at Point Xlies on What are thecoordinates of X such that the ratio PX:XQ is 3:2? Show your work or explain your answer.

Part EA new circle is drawn with the diameter of Explain whether or not this new circle is similar to the givencircle. If the circles are similar, include the ratio of dilation in your explanation.



4. A town has a park in the shape of a right triangle, formed by Main Street, Pecan Street, and Avenue B.

Part ADetermine the length of the path to the nearest foot. Show your work.

Part BUsing the path length you found in Part A, determine the distance along the section of Avenue B that connects the path toMain Street to the nearest foot. Show your work.

Part CDetermine the area of the park. Show your work. Round your answer to the nearest 100 square feet.



Part DWhat is the relationship between the two triangles within the park that are separated by the path? Explain your answer usingappropriate postulates, theorems, or definitions.



5. When the point is rotated clockwise about the point the coordinates of the resultingpoint are

Part AIf the point is rotated clockwise about the point what are the coordinates of the resultingpoint? Show your work.

Part BProve that using the coordinates of points A, B, and B’.

Part CProve that using the coordinates of points A, B, and B’.





6. Golf balls have a diameter of approximately 4.25 centimeters (cm). Use 3.14 as an approximation of asneeded for the following parts.

Part ACompany M ships golf balls in cartons that have the least volume needed to hold 24 golf balls. Each cartonis in the shape of a rectangular prism. The base of a carton is 17 cm long and 8.5 cm wide. The largestcross section of the carton is a rectangle. What is the area of the rectangle? Give your answer to thenearest square centimeter.

Part BCompany N ships golf balls in cartons that hold twelve balls. Each carton measures 4.25 cm by 12.75 cmby 17 cm. What percent of the total volume of the carton do the golf balls fill? Round your answer to thenearest percent.

Part CCompany P wants to create a container that will take up less space. The supplier cuts a 4.5 cm by 12.75cm face from a carton and rotates it about the perpendicular bisector of its shorter side. The shapegenerated by the rotation is a cylinder. What is the volume of the cylinder? Round your answer to thenearest cubic centimeter.



7. A parabola has a focus at the point (6, 0), and the equation of the directrix is

Part ADetermine the vertex of the parabola. Explain your answer.

Part BProve that point (12, 8) is on the parabola. Show your work.

Part CThe point (x, y) is on the parabola. Use this information to derive the equation for the parabola. Show yourwork.

Part DSketch a graph of the parabola, indicating the vertex, focus, directrix, and two additional points on theparabola.





8. Circle O has radius 1. The diagram shows three regular n-gons inscribed in circle O.

Part AWithout performing the calculations, explain how to find the area of the n-gons pictured above.

Part BUse the method you described in Part A to calculate the area of a dodecagon and a 100-gon that can beinscribed in circle O. Show your work and round to the nearest hundredth.

Part CSuppose n keeps increasing without bound. What value do the areas of the inscribed n-gons approach?Explain your answer.

Part DUse the results you found in Parts A through C to make a generalization about the area of a circle.





9. The graph of the function is shown on the coordinate plane. The graph of another function, isalso shown. The function can be obtained by applying 3 transformations to

Part AWrite an equation to represent the function Explain your answer using the transformations applied to

Part BIf the graph of is transposed 2 units right and 2 units up, the result is the graph of What are thecoordinates of the vertex of Explain your answer.

Part CWhat is the value of xif Show your work or explain your answer.

Part DWhat is the interval on which the function is decreasing? Explain your answer.





10. The points and are the vertices of a triangle. Another point is given. Youmay use the coordinate plane to help answer the questions.

Part AProve that point is on Show your work.

Part BProve that is an altitude of and use this information to find the area of Show your work.

Part CIf is reflected across the y -axis, what is the slope of the image of side BC ? Explain your answer.



11. The coordinates of the vertices of are shown on a coordinate plane.



Part ATo the nearest tenth of a unit, what is the perimeter of Show your work.

Part BProve that triangle is a right triangle.

Part CAn altitude of is drawn from Mto What is the slope of the altitude? Show your work.



12. Point P is the endpoint of three rays, PQ , PR , and PS . The coordinates of the endpoints of the linesegments contained in the rays are shown in the graph.



For each question, use the information given in the graph.

Part ALucy plotted points on that divided the line segment into 5 equal segments. What are the coordinates ofthe point nearest to P ? Show work or explain your answer.

Part BTed plotted point T at the y-intercept of , dividing into line segments PT and TR . What is the ratio ofthe lengths of these two line segments? Show work or explain your answer.

Part CManuel divided into 4 equal line segments. What are the endpoints that Manuel had to plot? Show yourwork.



13. Figure RSTUhas the vertices of and

Part A



Does figure RSTU have a pair of parallel sides? Explain your answer.

Part BWhat is the best name for figure RSTU ? Explain your answer algebraically.

Part CIf figure RSTU is translated 3 units to the right and reflected over the x -axis, write two expressions todescribe how the given transformations affect the x - and y -coordinates of the figure. Explain your answer.



14. Figure RSTUVW is a regular hexagon.

Part AList all of the possible lines of reflection for figure RSTUVW .

Part BIn terms of degrees, describe 3 rotations that would carry the figure onto itself. Explain your answer.

Part CLuke says that a regular octagon has eight lines of reflection. Is he correct or incorrect? Explain youranswer.





15. Figure DEFrepresents of circle E .

Part ADescribe three transformations on figure DEF that could be performed to complete circle E .

Part BThe center of the circle Eis moved 5 units to the right and 8 units down. If represents a point on figureDEF , write an ordered pair with x and y that represents the corresponding point in figure D’E’F’ after thetransformation of circle E. What are the coordinates of D’ and F’ in the transformed figure?



16. The vertices of a trapezoid are and

Part A



What are the coordinates of the vertices of trapezoid that is formed when ABCD is reflected throughthe y -axis?

Part BTrapezoid is then reflected through the line to form What are the coordinates of

Part CWhat transformation could be performed on to result in the original trapezoid ABCD ?

Part DParts A, B, and C give a sequence of three transformations that carry the original trapezoid ABCD back ontoitself. Rena claims that she obtained the same result (transforming ABCD back onto itself) using a differentset of three transformations.

●The second transformation is reflecting through the x -axis.●The third transformation is reflecting through the y-axis.

What was the first transformation that Rena did? Explain how you got your answer.



17. The points on the coordinate grid represent two cities. The city located at is in the Central StandardTime zone and the other city, located at is in the Eastern Standard Time zone.



Part AWhat is the distance between the two cities? Show your work.

Part B

If someone begins at and travels of the way toward they would arrive at the time zoneboundary.

Which point best represents the location of the time zone boundary? Show your work.



18. Antonio draws triangle RST on a coordinate grid.



Part AWhat are the lengths of the sides of triangle RST ? Show your work.

Part BAntonio states that RST is an isosceles triangle. Is Antonio correct? Use your answers from Part A toexplain your answer.

Part CAntonio creates triangle EFG congruent to triangle RST. He plots the points and What is thedistance between these two points?

Part DWhere would Antonio need to plot point G in order for the two triangles to be congruent? Show your workor explain your answer.



19. An architect created a floor plan of a home that will have a hexagonal living area and two squarebedrooms. The sketch of the floor plan shows hexagon PQRSTU and two congruent squares, UVWP and RYXQ.



In the diagram and

Round all of your answers to the parts to the nearest tenth of a foot.

Part AThe owner wants the architect to include a deck that attaches to segment TS. What is the length of

Explain your answer.

Part BA landscaper designed a garden with a decorative fence outside the home from point V to point T. What willbe the length of the fence? Explain your answer.

Part COne part of the electrical wiring will connect from point P to some point on What is the shortest lengththat the wiring can be? Explain your answer.



20. A circle can be circumscribed about a triangle or inscribed within a triangle.

Part A



Explain how to inscribe a circle in a triangle.

Part BHow does the construction of a circle circumscribed about a triangle compare to your explanation for PartA? Explain your answer.



21. The equations of two circles are shown below.

Part AWhat are the center and the radius of the circle for the equation Show and explainyour work.

Part BWhat must be true about the value of k in order for to represent a circle?Explain your answer.





22. Answer each part regarding a figure in the coordinate plane.

Part AQuadrilateral JKLM is defined by the points and Prove whether thequadrilateral defined by the four given points is a parallelogram. Show and explain your work.

Part BA circle is centered at point K and has a radius of 3 units. Determine whether the point lies on thecircle. Show and explain your work.



23. Lila was given and as shown.

She was also told that and Lila successfully showed that and are similar,using a sequence of transformations.

Part A



The following diagram shows Lila's first step.

Explain the transformation Lila used to create from

Part BThe following diagram shows Lila's second step.

Explain the transformation Lila used to create from

Part CExplain why the steps Lila used established that and are similar.

Part DSuppose Lila had switched the order of her transformations. Explain how that sequence of transformationswould be done and whether the change would still establish that and are similar.





24. Irene tried to explain to her classmates the formula for the area of a circle. The picture shows the circleshe constructed that has a radius, r, and 16 congruent sectors.

The following picture shows how Irene cut and arranged the 16 sectors.

Irene told her classmates to think of the 16 sectors as fitting into a rectangle that has a length, L, andwidth, r. All the dimensions are in units.

Part AWhat does L represent in terms of r? Explain your answer.

Part BHow does the area of the rectangle compare to the area of the shaded sectors? Explain your answer.

Part CUsing your results from Part A and Part B, what is the area of the shaded portion of the rectangle in termsof r? Show your work.

Part DIrene was trying to give her classmates a concrete example of how to derive the formula for the area of acircle. How could she have improved her approach? Explain your answer.





25. Sphere S is placed inside a right cone so that point R on the sphere is the center of the base of thecone. Point V represents the vertex of the cone. The sphere is tangent to the cone at points X and Y. PointsR, S, and V are collinear. The diagram represents this situation.

Part AWhat is the radius, in centimeters, of the sphere? Show your work and explain your answer.

Part BWhat is the percentage of the volume of the cone occupied by the sphere? Show your work and explainyour answer.

Part CThe cone is cut by a plane that is parallel to its base. Point S is the center of the base of the new figure asshown.

What is the volume of the new figure in terms of Show your work and explain your answer.





26. In the given figure, and are isosceles triangles.

Part Aand are proven congruent by the given information. However, does there exist a transformation

that could also prove these triangles are congruent? If so, state that transformation and explain yourreasoning.

Part BThe measures of and what are the measures of the other angles in the figure?Show your work and explain your answers.

Part CWhen is reflected, translated, and rotated to create explain why the triangles would becongruent.



27. In the figure shown, and are right triangles.



Part AIdentify two sets of similar triangles that can be determined by the Angle-Angle Similarity Postulate.Explain your reasoning.

Part BUsing a two-column proof and your answer to Part A, prove the Pythagorean theorem for

Part CGiven that cm, cm, and cm, determine the length of in centimeters. Show andexplain your work.



28. Hema was asked to derive the formula for the area of a sector. She drew circle K that has a radius of r units. She shaded and labeled the area of sector JKL that has a central angle of The diagram shows heroriginal circle K and then the circle with sector JKL.

Part AHema used a ratio to compare the area of sector JKL to the area of the circle. If she used X to represent



the area of sector JKL, what is a ratio that shows this comparison? Explain your answer.

Part BHema used another ratio to compare the measure of angle JKL to What is a ratio that shows thiscomparison?

Part CWrite a proportion that uses the two ratios you wrote in Part A and Part B. Explain why your proportion isvalid.

Part DFrom your answer in Part C, write the formula for X, the area of sector JKL. Explain your answer by statingthe algebraic property that allows you to write your formula.



29. In the figure shown, is a right triangle and

Explain how to determine

Type your answer in the box.You may use numbers, symbols, and/or text in your answer.



30. An antenna has a support wire that is anchored to the ground 20 feet from the base of the antenna onlevel ground. The support wire is taut and reaches the top of the antenna. The angle of elevation from theground to the top of the antenna is as shown in the figure.

What is the difference between the length of the support wire and the height of the antenna? Round yourfinal answer to the nearest tenth of a foot.


31. Serena drew two circles, as shown below. The radii of the circles are given in centimeters. The length ofa chord of the smaller circle is centimeters. The length of the corresponding chord of the larger



circle is centimeters.

What is the scale factor between these circles that proves that they are similar? Write your answer as afraction in simplest form.


32. The coordinate plane shows and that and have the same slope.

Explain how the information given determines that and are similar.

Type your answer in the box.



You may use numbers, symbols, and/or text in your answer.

33. Explain whether or not the point lies on a parabola that has a focus of and a directrix


34. In the diagram shown,



Explain why is isosceles.


35. Leon is proving a theorem about a parallelogram using coordinate geometry. Three of the vertices of hisparallelogram are labeled in the coordinate plane shown.

What should be the coordinates of the fourth vertex for Leon's parallelogram in terms of a, b, and c?




36. Triangle RST is graphed on the coordinate plane.

If is rotated about the origin, what percent of the image lies in Quadrant I? Round your answer tothe nearest tenth of a percent.


37. A floor plan for the first floor of a school dormitory is shown.



All of the rooms will have doors that open to the inside of the building, and each floor will have a commonarea for the students to study in groups. There will be 4 entrances on each floor. Each entrance has a widthof 5 feet and will be located on the front and back of the building leading into the common area.

Part AEach dormitory room needs to be 29 feet by 20 feet and must be located along each outer wall of thedormitory so that each room can have a window. With these restrictions, and maximizing the number ofrooms on the floor, how many rooms will be on this floor? Explain your reasoning.

Part BThe area shaded in the floor plan represents the common area of the floor. Determine the square footageof the common area on this floor. Show your work.

Part CTo accommodate more rooms on the second floor and still keep the restrictions, the size of the rooms willbe decreased to 25 feet by 20 feet with the 20-foot side being along the outside wall. When the number ofrooms is maximized, what is the increase in the number of rooms? Explain your reasoning.

Part DWhat is the change in the common area from the first floor to the second floor? Explain your answer.



38. The lampshade shown can be represented as a truncated cone. The lampshade represents of theentire cone. The lampshade has a height of 9.5 inches and a diameter of 12 inches.



Part AHow much material would be needed to cover the outside of this lampshade? Show your work or explainyour answer. Leave your answer in terms of

Part BWhat is the diameter of the top of the lampshade? Explain your work.

Part CAfter covering the lampshade, a trim will be placed around the upper and lower edges of the lampshade.How long will a piece of trim need to be in order to cover both edges? Show your work and round youranswer to the nearest tenth.

Part DA plane intersects the lampshade at a angle and does not intersect either base. What is the shape ofthe cross section that is produced by this plane?



39. Two cross-sections of a cylinder are circles. How are the cross-sections related to the bases of thecylinder?




40. Kylie made a cake consisting of ice cream and a chocolate crust. The cake is cylindrical and has adiameter of 10.5 inches. The ice cream filling, which is 10 inches in diameter, is surrounded on the sides,

top, and bottom by a chocolate crust that is thick. The height of the cake is inches.

What is the volume of the cake’s crust to the nearest hundredth?


41. Andrea has a bowling ball with a diameter of 8.5 inches and a density of 0.0497582 pounds per cubicinch. John has a bowling ball with a diameter of 5.6 inches and a density of 0.1413776 pounds per cubicinch. To the nearest tenth of a pound, how much heavier is Andrea's bowling ball than John's?




42. A carpenter is building stairs for an elevated deck. The three supporting pieces of the steps, called stringers, are each x inches wide as shown in the diagram.

An illustration of the side of one stringer is shown below.

All segments in the diagram are vertical or horizontal except for segment The length of every horizontalsegment is y inches, and the length of every vertical segment is z inches. Write a simplified equation thatgives the volume, V, of one stringer in terms of x, y, and z.




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