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1 GEOMETRY ADDITIONAL PRACTICE ITEMS Geometry Additional Practice Items This section has two parts. The first part is a set of 4 sample items for Geometry. The second part contains a table that shows for each item the standard assessed, the DOK level, the correct answer (key), and a rationale/explanation about the key and distractors. The sample items can be utilized as a mini-test to familiarize students with the item formats found on the assessment. All example and sample items contained in this guide are the property of the Georgia Department of Education. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 47 of 8

2 Geometry Formula Sheet You can find mathematics formula sheets on the Georgia Milestones webpage at and-assessment/assessment/pages/georgia- Milestones-Assessment-System.aspx. Look under EOC Resources. Page 48 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

3 Item Selected-Response Look at the triangle Which triangle is similar to the given triangle? 5 A. 77 B C D. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 49 of 8

4 Item Constructed-Response The following are the steps to construct an equilateral triangle. Determine the error in the steps. Write your answer on the lines provided. C C A B A B A B Page 50 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

5 Item 3 Extended Constructed-Response Right ABC with altitude BD. B A D C Prove ABC is similar to BDC. Statement Reason Item 4 Technology-Enhanced Triangle ABC is similar but not congruent to triangle DEF. Part A Which series of transformations could map triangle ABC onto triangle DEF? A. translation 4 units up, rotation 75 about the origin B. reflection across the line y =, rotation 90 about the origin C. translation 3 units left, dilation of scale factor centered at the origin D. reflection across the line x =, reflection across the line y = 5 Part B Which equation must be true about triangle ABC and triangle DEF? A. AB = DE B. AC = EF C. m/a + m/b = m/d + m/f D. m/a + m/c = m/d + m/f Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 5 of 8

6 Item 5 Selected-Response Which equation is true? A. sin40 = tan50 B. cos40 = cos50 C. sin40 = sin50 D. cos40 = sin50 Item 6 Technology-Enhanced Triangle GHJ is a right triangle. Angle G has a measure of g, angle H has a measure of h,and angle J is a right angle. Part A Select TWO equations that must be true. A. sin(h ) = sin( g ) B. cos( g ) = sin( h ) C. cos(h ) = cos( g ) D. sin(h ) + cos( h ) = sin( g ) + cos( g ) E. sin( g ) + cos( h ) = cos( g ) + sin( h ) Part B sin( g ) Given that tan( g ) =, which ratio must have a value equivalent to the cos( g ) tangent of g? cos( h ) A. sin( g ) cos( h ) B. sin( h ) sin( h ) C. cos( h ) sin( h ) D. cos( g ) Page 5 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

7 Item 7 Selected-Response Which point is NOT on a circle with a center of (0, 0) and a radius of 0? A. (0, 5) B. (0, 0) C. (0, 0) D. ( 8, 6) Geometry Additional Practice Items Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 53 of 8

8 Item 8 Constructed-Response Study the triangle. x 5 in. 3 in. Explain how you can determine the value of sin x. Use the word theta in your explanation instead of the symbol. Write your answer on the lines provided. Page 54 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

9 Item 9 Constructed-Response πr Explain why the formula for the area of a sector is A = 360, where r is the radius of the circle and θ is the measure in degrees of the central angle of the sector. Use the word pi in your explanation instead of the symbol π. Write your answer on the lines provided. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 55 of 8

10 Item 0 Technology-Enhanced The figure shows circle C with tangent lines QR and SR. Q R C x S P The measure of /QCS is x. Select THREE statements that are true about the figure. A. The measure of /QPS is (90 x). B. The measure of /QPS is x. C. The measure of /PSR is 90. D. The measure of /CQR is 90. E. The measure of /QRS is (80 x). F. The measure of /QRS is x. Page 56 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

11 Item Selected-Response Points A, B, C, D, and E are located on the circle O, as shown in this figure. A E O x B D C The measure of CD is 80. What is the value of x? A. 50 B. 40 C. 35 D. 5 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 57 of 8

12 Item Constructed-Response A pyramid and a rectangular prism have congruent bases and equal heights. Write a statement comparing the volume of the figures, and explain your reasoning. Write your answer on the lines provided. Page 58 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

13 Item 3 Selected-Response What is the sequence of transformations that carry triangle ABC to triangle QRS? y 6 4 B A C R x -4 Q -6 S A. Triangle ABC is reflected across the line x = 3. Then it is translated units down. B. Triangle ABC is reflected across the line x = 3. Then it is translated 6 units down. C. Triangle ABC is translated units to the left. Then it is rotated 90 degrees counterclockwise about the point (, ). D. Triangle ABC is translated units to the right. Then it is rotated 90 degrees counterclockwise about the point (, ). Item 4 Selected-Response Which transformation on quadrilateral ABCD produces an image that does not preserve distance between points in quadrilateral ABCD? A. reflection across y = x B. translation 3 units down and 4 units to the right C. dilation by a scale factor of D. rotation of 70 degrees Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 59 of 8

14 Item 5 Selected-Response Look at quadrilateral QRST. y R (3, 6) S (8, 8) 4 Q (, 3) T (6, ) x What is the image of point R after a counterclockwise rotation of 70 degrees about the origin? A. (6, 3) B. ( 3, 6) C. ( 6, 3) D. (3, 6) Page 60 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

15 Item 6 Selected-Response Look at the square WXYZ on this coordinate plane. y 6 4 X W Y x -4 Z -6 What is the perimeter of the square WXYZ? A. 0 units B. 5.6 units C. 3 units D units Item 7 Selected-Response What is the coordinate of point P that lies along the directed line segment from Q(, 5) to R(7, ) and partitions the segment in the ratio of 3 to? A. (3, 4.) B. (4.5, 8.5) C. (5, 9.) D. (5, 7) Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 6 of 8

16 Item 8 Selected-Response What is the equation of a line that is perpendicular to y = x 6 and passes through the point (6, 4)? A. y = x+ B. y = x+7 C. y = x 8 D. y = x+6 Page 6 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

17 Item 9 Selected-Response Study this equation of a circle. x 6x + y + y + 6 = 0 Which of these represents the center and radius of the circle? A. center: (3, ), radius: 4 B. center: ( 3, ), radius: 4 C. center: (3, ), radius: D. center: ( 3, ), radius: Item 0 Selected-Response What proves that figure ABCD is a parallelogram? B A C A. Diagonal BD bisects angle ABC. B. Side AB is equal to diagonal AC. C. Diagonals BD and AC bisect one another. D. Diagonal BD is greater than diagonal AC. D Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 63 of 8

18 Item Constructed-Response One bag of lawn fertilizer can cover approximately 5,000 square feet. Mike s lawn is about 500 square feet. When Mike applies fertilizer to his lawn, he applies it to 3 4 of his lawn only. Part A: About how many complete times can Mike fertilize his lawn with one bag of fertilizer? Part B: Mike fertilizes his lawn an average of 4 times per year. About how many full years will he be able to fertilize his lawn with one bag of fertilizer? Item Constructed-Response A student draws a card from a standard deck and then draws another card without replacing the first card. Explain why the probability of picking an ace on the first draw and the probability of picking a 7 on the second draw are NOT independent events. Write your answer on the lines provided. Page 64 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

19 Item 3 Selected-Response Geometry Additional Practice Items When rolling a fair, six-sided number cube, what is the probability of rolling an even number or a number less than 3? A. 5 6 B. 3 C. D. 3 Item 4 Selected-Response What is the probability of rolling a 5 on a fair, six-sided number cube if you know that you rolled an odd number? A. 6 B. 3 C. D. 3 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 65 of 8

20 ADDITIONAL PRACTICE ITEMS ANSWER KEY Item Standard/ Element DOK Level Correct Answer Explanation MGSE9-.G.SRT.3 A MGSE9-.G.CO.3 N/A 3 MGSE9-.G.SRT.5 3 N/A The correct answer is choice (A) because when the third angle is found, corresponding angles are congruent. Choices (B), (C), and (D) are incorrect because they have angle measures that are different than the original triangle. See scoring rubric and exemplar responses on page 7. See scoring rubric and exemplar responses beginning on page 7. 4 GSE-: G.SRT.5 Part A: C Part B: D See scoring rubric on page MGSE9-.G.SRT.7 D The correct answer is choice (D) because the angles are complements, the sine of an angle is equal to the cosine of the angle s complement. Choices (A), (B), and (C) are incorrect because they do not correspond to any trigonometric identities. 6 GSE-: G.SRT.7 Part A: B/D Part B: B See scoring rubric on page MGSE9-.G.GPE.4 D 8 MGSE9-.G.SRT.8 N/A 9 MGSE9-.G.C.5 N/A The correct answer is choice (A) because the point (0, 5) is only 5 units away from the center of the circle. Choices (B), (C), and (D) are incorrect because they are 0 units away from the center of the circle. See scoring rubric and exemplar responses on page 76. See scoring rubric and exemplar responses on page GSE-: G.C. 3 B/D/E See scoring rubric on page 78. Page 66 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

22 Item Standard/ Element DOK Level Correct Answer 6 MGSE9-.G.GPE.7 B 7 MGSE9-.G.GPE.6 C 8 MSGE9-.G.GPE.5 D Explanation The correct answer is choice (B) 5.6 units. Apply the distance formula to find the length of one side, which is 6.4. Since this is a square, multiply 6.4 by 4 to obtain the perimeter. Choice (A) is incorrect because the number of unit squares on a line segment were counted to estimate the length and then multiplied by 4. Choice (C) is incorrect because the number of unit squares across the square were counted to estimate the length and then multiplied by. Choice (D) is incorrect because it is the approximate area of the square. The correct answer is choice (C) (5, 9.). The rise from Q to R is 5 and the run from Q to R is 7. Multiply each value by the ratio 3 5 and then add that amount to the original coordinates (starting point) to find the x- and y-values for point P. Choice (A) is incorrect because these are the x- and y-values that need to be added to point Q. Choice (B) is incorrect because this is the midpoint of the line segment QR. Choice (D) is incorrect because this is the difference of the x-coordinates and y-coordinates. The correct answer is choice (D). This response is correct because the negative reciprocal of the slope, along with the x- and y-values of the given point, is substituted into the pointslope formula to find the y-intercept. Choice (A) is incorrect because this line is parallel, not perpendicular to the given line. Choices (B) and (C) are incorrect because the negative reciprocal is not used to find the equation of the perpendicular line. Page 68 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

23 Item Standard/ Element DOK Level Correct Answer 9 MGSE9-.G.GPE. C 0 MGSE9-.G.CO. C MGSE9-.G.MG. 3 N/A MGSE9-.S.CP.5 N/A 3 MGSE9-.S.CP.7 B Explanation The correct answer is choice (C) because when the equation is changed to standard form using completing the square, the h- and k-values are 3 and and r = 4, so r =. Choices (A) and (B) are incorrect because the radius comes from taking the square root of the constant in standard form. Choice (D) is incorrect because the signs of the center are opposite. The correct answer is choice (C) because if diagonals bisect one another, then the quadrilateral is a parallelogram. Choices (A), (B), and (D) are incorrect because there is not enough information to prove the quadrilateral is a parallelogram. See scoring rubric and exemplar responses on page 80. See scoring rubric and exemplar responses on 8. The correct answer is choice (B) because an even number or a number less than 3 includes the outcomes of,, 4, and 6 and there are 6 outcomes in the sample space; 4 6 simplifies to. Choice (A) is incorrect because there is 3 an overlap of in the outcomes of an even number and a number less than 3, and the overlap must be subtracted: P(A or B) = P(A) + P(B) P(A B). Choice (C) is incorrect because it is the probability of an even number only. Choice (D) is incorrect because it is the probability of a number less than 3 only. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 69 of 8

24 Item Standard/ Element DOK Level Correct Answer 4 MGSE9-.S.CP.3 B Explanation The correct answer is choice (B) because with the conditional probability, we assume that an odd number was rolled, which reduces our sample space to, 3, and 5. Out of those possibilities, the probability of rolling a 5 is 3, successful outcome out of 3 total outcomes. Choice (A) is incorrect because it is the probability of rolling 5 without knowing an odd number was rolled. Choice (C) is incorrect because it is the probability of rolling an odd number. Choice (D) is incorrect because it is the complement of the correct answer. Page 70 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

25 ADDITIONAL PRACTICE ITEMS SCORING RUBRICS AND EXEMPLAR RESPONSES Geometry Additional Practice Items Item Scoring Rubric Points 0 Description Student demonstrates complete understanding of constructing an equilateral triangle. Award points for a student response that contains both of the following elements: States that in step, line segment BC is drawn before the necessary arcs are drawn. States that point C is formed from the intersection of the two arcs drawn from the endpoints A and B. Student shows partial understanding of constructing an equilateral triangle. Award point for a student response that contains only one of the following elements: States that in step, line segment BC is drawn before the necessary arcs are drawn. States that point C is formed from the intersection of the two arcs drawn from the endpoints A and B. Student demonstrates limited to no understanding of constructing an equilateral triangle. Exemplar Response Points Awarded Sample Response In step, line segment BC cannot be drawn until after both arcs are drawn from endpoints A and B. Point C is the intersection of the arcs drawn from endpoints A and B. The segments BC and AC are drawn after that intersection is found. The error is in step because line segment BC was drawn too early. 0 Student does not produce a correct response or a correct process. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 7 of 8

26 Item 3 Scoring Rubric Points Description Response demonstrates a complete understanding of geometric proofs. Give 4 points for a complete proof with justifications for each step. Scoring Note: There are multiple valid ways of solving. Accept any valid method. Response demonstrates a nearly complete understanding of geometric proofs. Give 3 points for any of the following response types: Incomplete proof with one step missing. Incomplete proof with error in a statement or justification. Scoring Note: There are multiple valid ways of solving. Accept any valid method. Response demonstrates a partial understanding of geometric proofs. Give points for any of the following response types: Incomplete proof with steps missing. Incomplete proof with errors in a statement or justification. Incomplete proof with step missing and error in a statement or justification. Scoring Note: There are multiple valid ways of solving. Accept any valid method. Response demonstrates a minimal understanding of geometric proofs. Give point for 3 errors: Steps missing. Error/s in a statement or justification. Scoring Note: There are multiple valid ways of solving. Accept any valid method. Response demonstrates limited to no understanding of geometric proofs. Page 7 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

27 Item 3 Exemplar Response Points Awarded 4 Sample Response Step Statement Reason ABC is a right triangle Given Altitude BD Given 3 BDC is a right angle Definition of Altitude 4 ABC BDC All right angles are congruent. 5 BCD BCA Reflexive Property of Congruence 6 ABC BDC Angle-Angle (AA) Similarity 3 Step Statement Reason ABC is a right triangle Given Altitude BD Given 3 BDC is a right angle Definition of Altitude 4 ABC BDC All right angles are congruent. 5 BCD BCA Complimentary Angles are Congruent 6 ABC BDC Angle-Angle (AA) Similarity Step Statement Reason ABC is a right triangle Given Altitude BD Given 3 BDC is a right angle Definition of Altitude 4 ABC BDC All right angles are congruent. 5 BCD BCA Complimentary Angles are Congruent 6 ABC ADB Angle-Angle (AA) Similarity Step Statement Reason ABC is a right triangle Given Altitude BD Given 3 ABC BDC All right angles are congruent. 4 ABC BDC Angle-Angle (AA) Similarity 0 Student does not produce a correct response or a correct process. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 73 of 8

28 Item 4 Scoring Rubric Points 0 Description A score of indicates complete understanding of how to use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. The student determines that the correct answer for Part A is Choice (C). AND The student determines that the correct answer for Part B is Choice (D). A score of indicates a partial understanding of how to use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. The student determines that the correct answer for Part A is Choice (C). OR The student determines that the correct answer for Part B is Choice (D). A score of 0 indicates limited to no understanding of how to use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Page 74 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

29 Item 6 Scoring Rubric Points Description A score of indicates a complete understanding of how to explain and use the relationship between the sine and cosine of complementary angles. The student determines that the correct answers for Part A are Choice (B) and Choice (D). AND The student determines that the correct answer for Part B is Choice (B). A score of indicates a partial understanding of how to explain and use the relationship between the sine and cosine of complementary angles. 0 The student determines that the correct answers for Part A are Choice (B) and Choice (D). OR The student determines that the correct answer for Part B is Choice (B). A score of 0 indicates limited to no understanding of how to explain and use the relationship between the sine and cosine of complementary angles. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 75 of 8

30 Item 8 Scoring Rubric Points 0 Description Student demonstrates complete understanding of finding the sine of an angle in a triangle. Award points for a student response that contains both of the following elements: Finds the length of the third side. Explains how to find the sine of the angle. Student shows partial understanding of finding the sine of an angle in a triangle. Award point for a student response that contains only one of the following elements: Finds the length of the third side. Explains how to find the sine of the angle. Student demonstrates limited to no understanding of finding the sine of an angle in a triangle. Exemplar Response Points Awarded Sample Response Use the Pythagorean theorem to find that the length of the third side of the triangle is 4 inches. The sine of an angle is the ratio of the length of the opposite leg compared to the length of the hypotenuse. So, the sine of theta is equal to the ratio of 4 to 5. Use the Pythagorean theorem to find that the length of the third side of the triangle is 4 inches. The sine of an angle is the ratio of the length of the opposite leg compared to the length of the adjacent leg. So, the sine of theta is equal to the ratio of 4 to 3. 0 Student does not produce a correct response or a correct process. Page 76 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

31 Item 9 Scoring Rubric Points 0 Description Student demonstrates complete understanding of the equation for the area of the sector. Award points for a student response that contains both of the following elements: Explains the meaning of π 360. Explains the meaning of πr. Student shows partial understanding of the equation for the area of the sector. Award point for a student response that contains only one of the following elements: Explains the meaning of π 360. Explains the meaning of πr. Student demonstrates limited to no understanding of the equation for the area of the sector. Exemplar Response Points Awarded Sample Response The first part of the formula shows the degree measure of the central angle of the sector divided by the degree measure of the entire circle. So, it represents the fraction of the circle that consists of the sector. The second part of the formula shows pi times the radius squared, which is the area of the entire circle. So, the product of these two parts represents the fraction of the area of the circle that is included in the sector. The fraction in the formula shows the degree measure of the central angle of the sector divided by the degree measure of the entire circle. So, it represents the fraction of the circle that consists of the sector. 0 Student does not produce a correct response or a correct process. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 77 of 8

33 Item Scoring Rubric Points 0 Description Student demonstrates complete understanding of the formulas for the volume of pyramids and rectangular prisms. Award points for a student response that contains both of the following elements: Explains that the volume of a pyramid is equal to of the area of its base 3 times its height and the volume of a rectangular prism is equal to the area of its base times its height. Explains that the volume of the pyramid in the problem is the volume of the 3 rectangular prism (or the volume of the rectangular prism is triple the volume of the pyramid) because the pyramid and the prism have the same height and bases with the same area. Student shows partial understanding of the formulas for the volume of pyramids and rectangular prisms. Award point for a student response that contains only one of the following elements: Explains that the formula for the volume of a pyramid is equal to of the area 3 of its base times its height and the volume of a rectangular prism is equal to the area of its base times its height. Explains that the volume of the pyramid in the problem is the volume of the 3 rectangular prism (or the volume of the rectangular prism is triple the volume of the pyramid) because the pyramid and the prism have the same height and bases with the same area. Student demonstrates limited to no understanding of the formulas for the volume of pyramids and rectangular prisms. Exemplar Response Points Awarded Sample Response The volume of a rectangular prism is equal to the area of its base times its height, and the volume of a pyramid is equal to one-third of the area of its base times its height. Since the pyramid and the rectangular prism have congruent bases, their bases have the same area. Since the heights are congruent, they also have the same height. So, the volume of the pyramid is one-third of the volume of the rectangular prism. The volume of a rectangular prism is equal to the area of its base times its height, and the volume of a pyramid is equal to one-third times the area of its base times its height. 0 Student does not produce a correct response or a correct process. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 79 of 8

34 Item Scoring Rubric Points 0 Description Student demonstrates a complete understanding of using the concepts of area and volume in a modeling situation. Award points for a student response that contains both of the following elements: 3 3 Student shows partial understanding of using the concepts of area and volume in a modeling situation. Award point for a student response that contains only one of the following elements: 3 3 Student demonstrates little to no understanding of using the concepts of area and volume in a modeling situation. Exemplar Response Points Awarded Sample Response Student does not produce a correct response or a correct process. Page 80 of 8 Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents

35 Item Scoring Rubric Points 0 Description Student demonstrates complete understanding of independent events. Award points for a student response that contains both of the following elements: Explains that the probability of picking an ace on the first draw affects the probability of picking a 7 on the second draw. Explains that without replacement, the sample space changes. Student shows partial understanding of independent events. Award point for a student response that contains only one of the following elements: Explains that the probability of picking an ace on the first draw affects the probability of picking a 7 on the second draw. Explains that without replacement, the sample space changes. Student demonstrates limited to no understanding of independent events. Exemplar Response Points Awarded Sample Response When the first card is drawn from the deck, there are 5 cards to choose from. Because the first card is not replaced, there are only 5 cards to choose from when the second card is drawn. So, the result of picking an ace on the first draw affects the sample space and, therefore, the probability of picking a 7 on the second draw. So, the events are not independent. The probability of choosing a card and not replacing it affects the probability of the second card drawn. 0 Student does not produce a correct response or a correct process. Georgia Milestones Geometry EOC Study/Resource Guide for Students and Parents Page 8 of 8

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