# Analytical Geometry- Common Core

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1 Analytical Geometry- Common Core 1. A B C is a dilation of triangle ABC by a scale factor of ½. The dilation is centered at the point ( 5, 5). Which statement below is true? A. AB = B C A B BC C. AB = B C BC A B B. AB = BC A B B C D. AB = A C BC B C 2. A B C is a dilation of triangle ABC by a scale factor of ½. The dilation is centered at the point (5, 5). What is the ratio of the area of A B C to the area of ABC? A. ½ B. 2 C. ¼ D In the coordinate plane segment M N is the result of a dilation of segment MN by a scale factor of ⅓. Which point is the center of dilation? A. (1, 3) C. ( 5, 0) B. (0, 0) D. ( 4, 1) 4. In the triangles shown ABC is dilated by a factor of ⅔ to form RST. Given that m B = 70 and m C = 50, what is the m R? A. 50 B. 60 C. 70 D In the triangles shown ABC is dilated by a factor of ⅔ to form RST. Given that TR = 6 cm, what is the length of CA? A. 4 cm B. 9 cm C. 4.5 cm D. 7.5 cm - 1 -

2 Analytical Geometry- Common Core 6. In the triangle shown, AB DE. What is the length of CD? A. 1.2 C. 6.0 B. 3.3 D In the triangle shown, HJ GK. The following shows a proof of the statement If a line is parallel to one side of a triangle and intersects the other two sides at distinct points, then it separates these sides into segments of proportional lengths. Which reason justifies Step 2? A. Alternate interior angles are congruent. B. Alternate exterior angles are congruent. C. Corresponding angles are congruent. D. Vertical angles are congruent. 8. Consider the triangles shown. Which can be used to prove the triangles are congruent? A. SAS B. SSA C. ASA D. SSS 9. Consider the triangles shown. Which would be a correct congruence statement? A. ABC DEF B. BCA EDF C. BAC DFE D. CAB FED - 2 -

3 Analytical Geometry- Common Core 10. In the diagram, AC DC. Which information would provide enough information to prove that ABC DEC? A. ED BA B. AD EB C. ED BA D. AD EB 11. In the diagram, STU is an isosceles triangle where ST is congruent to UT. The paragraph proof shows that S is congruent to U. It is given that ST is congruent to UT. Draw TV such that it bisects T. By the defintion of an angle bisector, STV is congruent to UTV. Also by the Reflexive Property, TV is congruent to TV. So, STV UTV by SAS. Thus, S is congruent to U by??. Which step is missing in the proof? diagram 1 diagram 2 A. Reflexive Property of Congruence B. Angle Congruence Postulate C. Definition of right angles D. CPCTC 12. Consider the construction of the angle bisector shown. Which could have been the first step in creating this construction? A. Place the compass point on point A and draw an arc inside Y. B. Place the compass point on point B and draw an arc inside Y. C. Place the compass point on vertex Y and draw an arc that intersects YX and. YZ D. Place the compass point on vertex Y and draw an arc that intersects point C. 13. Use the line segment HI to answer the question. Which step should be first to draw a line perpendicular to HI at midpoint J? A. Place the compass point on point H and set its width to less than HJ. B. Place the compass point on point H and set its width to more than HJ. C. Place the compass point on point J and set its width to less than HI. D. Place the compass point on point J and set its width to more than HI

4 Analytical Geometry- Common Core 14. Triangle ABC is shown. What is the value of cos A? A. 3 5 C. 4 5 B. 3 4 D There is a large tree 80 feet from a house. The owners are worried that the tree might hit their house if it fell and want to estimate the height of the tree. In the figure below, when the sun s angle of elevation is 50, the tree casts a shadow 80 feet long. Which can be used to find the height of the tree? A. sin 50 = 80 x C. cos 50 = 80 x B. tan 50 = x 80 D. sin 50 = x In right triangle HJK, J is a right angle and tan H = 1. Which statement about triangle HJK must be true? A. sin H = 1 2 B. sin H = 1 C. sin H = cos H D. sin H = 1 cos H 17. In right triangle ABC, A and B and are complementary angles. The value of cos A = What is the value of sin B? A B C D In the circle shown, BC is the diameter and m AB = 120. What is the measure of ABC? A. 15 B. 30 C. 60 D Circle P is dilated to form circle P. Which statement is ALWAYS true? A. The radius of circle P is equal to the radius of circle P. B. The length of any chord in circle P is greater than the length of any chord in circle P. C. The diameter of circle P is greater than the diameter of circle P. D. The ratio of the diameter to the circumference is the same for both circles

5 4.5 in. Analytical Geometry- Common Core 20. Circle with center P has tangents XY and ZY and chords WX and, WZ as shown in the figure. The measure of ZWX = 70. What is the measure, in degrees, of XYZ? A. 20 B. 35 C. 40 D Circle with center E is shown. The measure of CED = 145 and the length of CE is 12 cm. What is the length of CD? A. 29 π cm 72 B. 29 π cm 6 C. 29 π cm 3 D. 29 π cm Circle with center Y is shown. The measure of XYZ = 75 and the length of YZ is 9 cm. What is the area of the shaded part of the circle? A. 57 π cm2 4 B. 135 π cm2 8 C. 405 π cm2 8 D. 513 π cm The spokes of a bicycle wheel form 10 congruent central angles. The diameter of the circle formed by the outer edge of the wheel is 18 inches. What is the length, to the nearest 0.1 inch, of the outer edge of the wheel between two consecutive spokes? A. 1.8 inches B. 5.7 inches C inches D inches 24. What is the volume of a cylinder with a radius of 3 inches and a height of 9 2 inches? A B π in.3 π in.3 C. 27 π in.3 8 D. 9 4 π in.3-5 -

6 x cm x cm Analytical Geometry- Common Core 25. The surface area of the sphere with center P is 36π cm 2. What is the volume of the sphere? A. 12 π cm 3 C. 48π cm 3 P 3 in B. 36 π cm 3 D. 288 π cm The cone shown has a base with a radius of AB. The length of AB = 6 cm and the length of BC = 10 cm. What is the volume of the cone? A. 288 π cm 3 B. 360 π cm 3 C π cm 3 D. 96 π cm Two different cylinders are shown. Both cylinders have the same height, x cm, and same radius, y cm. The only difference is Cylinder B has been slanted by 60. Which statement below is true? A. Cylinder A has a bigger volume than Cylinder B B. Cylinder A has a smaller volume than Cylinder B C. Cylinder A has the same volume than Cylinder B Cylinder A y cm 60 y cm Cylinder B D. Not enough information is provided to determine which is cylinder has a bigger volume. 28. Which expression is equivalent to x x 3? A. x 5/2 B. x 5 C. 1 D. 1 x 5 x x - 6 -

7 Analytical Geometry- Common Core Diagnostic Test Which expression is equivalent to 27 12? A. 3 B. 2 6 C. 5 3 D Which expression is equivalent to x? 2 7 A x B. 4x C. 64x D x 31. Which expression has a value that is a rational number? A B. 3 ( ) C D Let a be a nonzero rational number and b be an irrational number. Which of these expressions MUST be a rational number? A. b + 0 B. a + a C. a + b D. b + b 33. A model of a house is shown. What is the perimeter, in units, of the model? A. 32x + 12 B. 46x + 25 C. 50x + 11 D. 64x Kelly makes two different-sized ceramic tiles in the shape of right isosceles triangles. This diagram shows the leg lengths of the small tile. Kelly makes a larger tile by increasing the length of each leg of the small tile by x inches. Which expression represents the length, in inches, of the hypotenuse of the large tile? A x B. (x + 3) 2-2 -

8 Analytical Geometry- Common Core Diagnostic Test -1 C. (x + 3) 2 D x 35. Which has the same value as i 9 + i 3? A. 2i B. 2 C. 2 D. 2i 36. Which has the same value as ( 5 3i)( 4 + 2i)? A. 26 2i B i C. 14 2i D i 37. What are the solutions to the equation 2x 2 + 3x + 9 = 0? A. x = 3 4 ± 21 4 i B. x = 3 4 ± 21 4 i C. x = 3 4 ± 3i 7 4 D. x = 3 4 ± 3i Given the expression 2n 2 n 3, what is the coefficient of the n term? A. 1 B. 1 C. 2 D Which expression is equivalent to 121x 4 64y 6? A. (11x 2 16y 2 )(11x y 2 ) B. (11x 2 16y 3 )(11x 2 16y 3 ) C. (11x 2 + 8y 2 )(11x 2 + 8y 2 ) D. (11x 2 + 8y 3 )(11x 2 8y 3 ) 40. Given the expression 6a 2 + 7a 3, which of the following is a factor of the expression? - 8 -

9 Analytical Geometry- Common Core Diagnostic Test -1 A. (3a 1) B. (2a 3) C. (6a + 1) D. (2a 1) 41. What is the vertex of the graph of f(x) = x x 9? A. (5, 66) B. (5, 9) C. ( 5, 9) D. ( 5, 34) 42. Consider the parabola in standard form y = 2x x 5. This is to be rewritten in vertex form by completing the square. In completing the square what should a be in the following equation so that it is still the same parabola y = 2(x 2 6x + 9) + a 5. A. 9 B. 9 C. 18 D Determine the equation of the parabola with a focus at (-2,5) and a diretrix of y=1. A. (x + 2) 2 = 8(y 3) B. (x 2) 2 = 8(y 3) C. (x + 2) 2 = 8(y + 3) D. (x 2) 2 = 8(y + 3) 44. The expression x x 600 represents a company s profit for selling x items. For which number(s) of items sold is the company s profit equal to \$0? A. 0 items B. 35 items C. 10 items & 60 items D. 20 items & 30 items 45. A garden measuring 8 feet by 12 feet will have a walkway around it. The walkway has a uniform width, and the area covered by the garden and the walkway is 192 square feet. What is the width of the walkway? A. 2 feet B. 3.5 feet - 8 -

10 Analytical Geometry- Common Core Diagnostic Test -1 C. 4 feet D. 6 feet 46. The formula for the surface area of a cone is SA = πr 2 + πrs. Which equation shows the formula in terms of s? A. s = SA πr πr2 B. s = SA πr + πr2 C. s = D. s = SA πr2 πr SA + πr2 πr 47. What are the solutions to the equation 2x 2 2x = 12? A. x = 4, 3 B. x = 3, 4 C. x = 2, 3 D. x = 6, What are the solutions of this system of equations? y = 5x 2 + 7x 6 y = 12x 6 A. (0, 6) and (1, 6) B. (0, 6) and (1, 6) C. (0, 6) and (1, 6) D. ( 6, 0) and (6, 1) 49. Use the table to answer the question. What is the average rate of change of f(x) over the interval 2 f(x) 0? A. 10 B. 5 C. 5 D What is the end behavior of the graph of f(x) = 0.25x 2 2x + 1? A. As x increases, f(x) increases. As x decreases, f(x) decreases. B. As x increases, f(x) decreases. As x decreases, f(x) decreases. C. As x increases, f(x) increases. As x decreases, f(x) increases. D. As x increases, f(x) decreases. As x decreases, f(x) increases

11 Analytical Geometry- Common Core Diagnostic Test Given that line m is parallel to line m, find the value of x that makes the diagram true (x+40) A. x = 10 B. x = 26 C. x = 30 D. x = 22 (4x+10) 52. The table shows a few values of each function for different values of x. Which would you predict BEST describes the comparison of the function values for f(x) and g(x)? A. The values of f(x) will always exceed the values g(x). B. The values of g(x) will always exceed the values f(x). x f(x) g(x) C. The values of f(x) exceeds the values g(x) over the entire interval [0, 5]. D. The values of g(x) exceeds the values f(x) over the entire interval [5, 6]. 53. What is the center of the circle given by the equation x 2 + y 2 10x 11 = 0? A. (5, 0) B. (0, 5) C. ( 5, 0) D. (0, 5) 54. Which shows an equation for a parabola with a focus at (4, 5) and a directrix of y = 1? A. y + 3 = 1 (x 4)2 4 C. y + 3 = 1 (x 4)2 8 B. y 4 = 1 (x + 3)2 4 D. y 4 = 1 (x + 3) A circle is centered at the origin and has a radius of 3 units. A horizontal line passes through the point (0, 3). In how many places does the line intersect the circle? - 8 -

12 Analytical Geometry- Common Core Diagnostic Test -1 A. 0 B. 1 C. 2 D. Infinitely many 56. Which statement BEST describes the graph of f(x + 6)? A. The graph of f(x) shifted up 6 units. B. The graph of f(x) shifted left 6 units. C. The graph of f(x) shifted right 6 units. D. The graph of f(x) shifted down 6 units. 57. If all of the medians of a triangle were constructed, what would the point where they all intersect be called? A. Incenter B. Orthocenter C. Centroid D. Circumcenter 58. Which students would be members of the group (A B)? A. {Amber, Chris} C. {Dave} Group A Group B B. {Bill} D. {Amber, Chris, Dave} Amber Bill Chris Dave 59. In a particular state, the first character on a license plate is always a letter. The last character is always a digit from 0 to 9. Let V represents the set of all license plates beginning with a vowel, and O represents the set of all license plates that end with an odd number. Which might be a license plate that belongs to the set (V O)? A. B. C. D. 60. A random survey was conducted about gender and hair color. This table records the data. What is the probability that a randomly selected person has blonde hair, give that the person selected is male? A B

13 Analytical Geometry- Common Core C D Assume that the following events are independent: The probability that a high school senior will go to college is 0.72 The probability that a high school senior will go to college and live on campus is 0.46 What is the probability that a high school senior will live on campus, given that the person will go to college? A B C D Each letter of the alphabet is written on a card using a red ink pen and placed in a container. Each letter of the alphabet is also written on a card using a black pen and placed in the same container. A single card is drawn at random from the container. What is the probability that the card has a letter written in red ink, the letter B, or the letter R? A. 1 2 B C D

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