Important Facts to Remember

Transcription

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2 Important Facts to Remember The sum of all angles inside any triangle equals 180 degrees. A linear pair equals 180 degrees. Vertical angles are congruent. θ (theta) means angle (used in formulas). Alternate interior angles are congruent. Same-side interior angles are supplementary. Alternate exterior angles are congruent. Same-side exterior angles are supplementary. Corresponding angles are congruent. Supplementary means 180 degrees. Complementary means 90 degrees. 2

3 Right angles are 90 degrees. Pythagorean Theorem can be used to find distances and the length of a side of a right triangle. Never try to use the Pythagorean Theorem to find the measure of an angle. ǁ means parallel (EX: AB ǁ CD) Bisect means to cut in half. Perpendicular means two lines cross to form a 90 degree right angle. Parallel means two lines never intersect. ~ means is similar to (EX: ΔABC ~ ΔDEF) means is congruent to (EX: ΔABC ΔDEF) is the symbol for angle. The diameter of a circle is the distance all the way across the circle. The radius of a circle is equal to half the length of its diameter. The perimeter of any figure it the sum of all of its sides. The perimeter of a circle is the same thing as the circle s circumference. The hypotenuse of a right triangle is the longest side of the right triangle. It s also the side directly across from the right angle. Equilateral triangles have all congruent sides and all congruent angles. Isosceles triangles have two congruent sides and two congruent angles. An angle of depression faces the ground. An angle of elevation faces the sky. An image is the result of a figure after a transformation has taken place. A pre-image is a figure that has not yet been transformed. The scale factor of a dilation is image/pre-image. A dilation is the only transformation that does not result in a congruent figure. A dilation either shrinks or enlarges a figure. Dilations result in similar figures. Congruent figures have all corresponding sides and angles equal to each other. Similar figures have congruent corresponding angles and proportional sides. The reflexive property of congruence makes a line segment or an angle congruent to itself. CPCTC means congruent parts of congruent triangles are congruent. CSSTP means corresponding sides of similar triangles are proportional. The triangle congruence theorems are: SSS, SAS, ASA, AAS, and HL. The triangle similarity postulates are: SAS, ASA, and AA. A rigid transformation is a transformation that results in a congruent figure. (EX: translations, reflections, and rotations) The transitive property: If two segments (or angles) are each congruent to a third segment (or angle), then they're congruent to each other. An inscribed angle is an angle whose vertex is on a circle. A central angle is an angle whose vertex is at the center of a circle. means is perpendicular to (EX: AB CD) 3

4 GSE Geometry EOC Study Guide 1. Mr. Rose is remodeling his house by adding a room to one side, as shown in the diagram below. In order to determine the length of the boards he needs for the roof of the room, he must determine the length from point A to point D. 3. A tackle shop and restaurant are on the shore of a lake and are 32 meters apart. A boat on the lake heading toward the tackle shop is a distance of 77 meters from the tackle shop. This situation is shown in the diagram below, where point T represents the location of the tackle shop, point R represents the location of the restaurant, point B represents the location of the boat. What is the length, to the nearest tenth of a foot, of AD? 2. A city is planning to replace one of its water storage tanks with a larger one. The city s old tank is a circular cylinder with a radius of 12 feet and a volume of 10,000 cubic feet. The new tank is a circular cylinder with a radius of 15 feet and the same height as the old tank. What is the maximum number of cubic feet of water the new tank will hold? The driver of the boat wants to change direction to head toward the restaurant. Which of the following is closest to the value of x? 4. Circle Q has a radius of 5 units with center Q (3.7, -2). Which of the following equations defines circle Q? A. (x ) 2 + (y 2) 2 = 5 B. (x ) 2 + (y 2) 2 = 25 C. (x 3. 7) 2 + (y + 2) 2 = 5 D. (x 3. 7) 2 + (y + 2) 2 = 25 4

5 5. For his mathematics assignment, Lamar must determine the conditions that will make quadrilateral ABCD, as shown below, a parallelogram. 7. Allison created an embroidery design of a stylized star emblem. The perimeter of the design is made by altering semi-circle and quarter-circle arcs. Each arc is formed from a circle with a 2 ½ inch diameter. There are 4 semi-circle and 4 quarter-circle arcs, as shown in the diagram below. Given that the m DAB = 40, which of the following statements will guarantee that ABCD is a parallelogram? A. m ADC + m DCB + m CBA + 40 = 360. B. m DCB = 40 ; m ABC = 140 C. m ABC + 40 = 180 D. m DCB = Zach inscribed kite ABCD in a circle, as shown below. To the nearest whole inch, what is the perimeter of Allison s design? A. 15 inches B. 20 inches C. 24 inches D. 31 inches If the measure of arc ADC is 255 in Zach s design, what is the measure, in degrees, of ADC? 8. Jessica created two right triangles. She started with ΔJKL and drew an altitude from point K to side JL. The diagram below shows ΔJKL and some of its measurements in centimeters. Based on the information in the diagram, what is the length of x to the nearest tenth of a centimeter? 5

6 9. In ΔABC, BD is an altitude. 11. Figure ABCD is a rhombus. The length of AE is (x + 5) units, and the length of EC is (2x 3) units. What is the length of BD? A. 1 B. 2 C. 3 D Four students are choreographing their dance routine for the high school talent show. The stage is rectangular and measures 15 yards by 10 yards. The stage is represented by the coordinate grid below. Three of the students Riley (R), Krista (K), and Juliana (J) graphed their starting positions, as shown below. Which statement best explains why the equation x + 5 = 2x 3 can be used to solve for x? A. All four sides of a rhombus are congruent B. Opposite sides of a rhombus are parallel C. Diagonals of a rhombus are perpendicular D. Diagonals of a rhombus bisect each other 12. Malik runs on the trails at the park. He normally runs 1 complete lap around trail ABCD. The length of each side of the trail ABCD is shown in meters in the diagram below. Let H represent Hannah s starting position on the stage. What should be the x-coordinate of point H so that RKJH is a parallelogram? If trail EFGH is similar to trail ABCD, what is the minimum distance, to the nearest whole meter, Malik would have to run to complete 1 lap around trail EFGH? 6

7 13. On the coordinate grid below, quadrilateral ABCD has vertices with integer coordinates. 15. A top view of downtown Rockford is shown on the grid below, with Granite Park represented by quadrilateral ABCD. The shape of a new park, Mica Park, will be similar to the shape of Granite Park. Vertices L and M will be plotted on the grid from quadrilateral JKLM, representing Mica Park. Quadrilateral QRST is similar to quadrilateral ABCD with point S located at (5, -1) and point T located at (-1, -1). Which of the following could be possible coordinates for point Q? A. (6, -4) B. (7, -7) C. (-3, -7) D. (-2, -4) 14. The circle below is centered at the origin and contains the point (-4, -2). Which of the coordinates for L and M could be vertices of JKLM so that the shape of Mica Park is similar to the shape of Granite Park? A. L(4, 4), M (4, 3) B. L(7, 1), M(6, 1) C. L(7, 6), M(6, 6) D. L(8, 4), M(8, 3) 16. In the figure below, AB is parallel to DC. Which of the following is closest to the length of the diameter of the circle? A B C D Which of the following statements about the figure must be true? A. m DAB + m ABC = 180 B. m DAB + m CDA = 180 C. BAD ADC D. ADC ABC 7

8 17. The owners of a water park want to build a scaled-down version of a popular tubular water slide for the children s section of the park. The side view of the water slide, labeled ABC, is shown below. 18. Highlands Park is located between two parallel streets, Walker Street and James Avenue. The park faces Walker Street and is bordered by two brick walls that intersect James Avenue at point C, as shown below. What is the measure, in degrees, of ACB, the angle formed by the park s two brick walls? Points A, B and C, shown above, are the corresponding points of the scaleddown slide. Which of the following would be closest to the coordinates of a new point C that will make slide A B C similar to slide ABC? A. (90, 20) B. (77, 20) C. (50, 20) D. (47, 20) 19. Given that ABC DEC and m E 23, find m ACB. 8

9 20. What is the missing reason for the proof? Given: Parallelogram ABCD with diagonal BD Prove: ABD CDB 22. The equation for line A is shown: y = - 2/3x 4 Line A and line B are perpendicular, and the point (-2, 1) lies on line B. Write an equation for line B. A. Reflexive Property of Congruence B. ASA C. Alternate Interior Angles Theorem D. SSS 21. On a coordinate grid, a grocery store is located at (3,0) and the hardware store is located at (4,3). If the hardware store is the midpoint between the grocery store and the pharmacy, what is the approximate distance from the hardware store to the pharmacy? (Note: 1 unit equals 5 miles) A. 7.5 miles B. 7.9 miles C miles D miles 23. The population of Florida in 2010 was 18,801,310 and the land area was 53,625 square miles. The population increased 5.8% by A. To the nearest whole number, what is the population density, in people per square mile, for Florida in 2014? B. To the nearest whole number, how much did the population density, in people per square mile, increase from 2010 to 2014? 9

10 24. Polygon ABCDE is shown on the coordinate grid. 26. At Buckroe Beach, Diego is using sand to make a circular mold with a rotationally symmetric star on top, as shown in the figure. How many degrees are in XY? What is the perimeter, to the nearest hundredth of a unit, of polygon ABCDE? 25. Points A, B, and C are collinear (meaning the points are on the same line) and AB:AC = 2/5. Point A is located at (-3, 6), point B is located at (n, q), and point C is located at (-3, -4). What are the values of n and q? (Warning: this problem is mean Read carefully!) 27. Mrs. Scott surveyed 240 men and 285 women about their vehicles. Of those surveyed 155 men and 70 women said they own a red vehicle. If a person is chosen at random from those surveyed, what is the probability of choosing a woman or a person that does NOT own a red vehicle? A. 14/57 B. 71/105 C. 74/105 D. 88/ Margret spins two spinners that have four equal sections numbered 1 through 4. If she spins a 4 on at least one spin, what is the probability that the sum of her two spins is an odd number? A. ¼ B. 7/16 C. 4/7 D. 11/16 10

11 29. Mrs. Scott surveyed 240 men and 285 women about their vehicles. Of those surveyed 155 men and 70 women said they own a red vehicle. If a person is chosen at random from those surveyed, what is the probability of choosing a woman or a person that does NOT own a red vehicle? A. 14/57 B. 71/105 C. 74/105 D. 88/ 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 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, given that the person selected is male? 32. 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. 33. Which students would be members of the group (A B)? A B C D A. {Amber, Chris} B. {Bill} C. {Dave} D. {Amber, Chris, Dave} 11

12 34. 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? A. 0 B. 1 C. 2 D. Infinitely many 37. A model of a house is shown. What is the perimeter, in units, of the model? 35. What is the center of the circle given by the equation below: A. 32x + 12 B. 46x + 25 C. 50x + 11 D. 64x + 24 x 2 + y 2 10x 11 = 0 A. (5, 0) B. (0, 5) C. (-5, 0) D. (0, -5) 36. Given that line m is parallel to line m, find the value of x that makes the diagram true. 38. 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. x = 10 B. x = 26 C. x = 30 D. x = 22 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 D. Not enough information is provided to determine which is cylinder has a bigger volume. 12

13 39. 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? 41. 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. A. 1.8 inches B. 5.7 inches C inches D inches 42. In the circle shown, BC is the diameter and the measure of arc AB = 120. What is the measure of ABC? 40. Circle with center Y is shown. The measure of WYZ = 75 and the length of YZ is 9 cm. What is the area of the shaded part of the circle? A. 15 B. 30 C. 60 D. 120 A. 57/4 π B. 135/8 π C. 405/8 π D. 513/8 π 43. In right triangle ABC, A and B are complementary angles. The value of cos A = 5/13. What is the value of sin B? A. 5/13 B. 12/13 C. 13/12 D. 13/5 13

14 Use the diagram below to answer questions 44 and In the triangle shown, HJ ǁ 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. 44. Consider the triangles shown. Which can be used to prove the triangles are congruent? A. SAS B. SSA C. ASA D. SSS 45. Consider the triangles shown. Which would be a correct congruence statement? A. B. C. D. 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. 14

15 47. In the triangle shown, AB ǁ DE. What is the length of CD? 50. In the coordinate plane, segment M N is the result of a dilation of segment MN by a scale factor of 1/3. Which point is the center of dilation? A. 1.2 B. 3.3 C. 6.0 D. 7.5 Use the diagram below to answer questions 48 and In the triangles shown, ΔABC is dilated by a factor of 2/3 to form ΔRST. Given that the m B = 70 and the m C = 50, what is the m R? A. 50 B. 60 C. 70 D Given that TR = 6 cm, what s the length of CA? A. 4 cm B. 9 cm C. 4.5 cm D. 7.5 cm A. (1, 3) B. (0, 0) C. (-5, 0) D. (-4, 1) 51. Which transformation results in a figure that is similar to the original figure but has a greater area? A. a dilation of QRS by a scale factor of 0.25 B. a dilation of QRS by a scale factor of 0.5 C. a dilation of QRS by a scale factor of 1 D. a dilation of QRS by a scale factor of Triangle ABC is given below. What is the value of cos A? 15

16 Use the diagram below to answer question 53 and Consider the triangles shown. Which can be used to prove the triangles are congruent? A. SSS B. ASA C. SAS D. AAS 53. ΔA B C is a dilation of ΔABC by a scale factor of ½. The dilation is centered at the point (5, 5). Which statement below is true? 56. A hot air balloon is 1,200 feet above the ground. The angle of depression from the basket of the hot air balloon to the base of a monument is 54. A. B. C. D. 54. What is the ratio of the area of ΔA B C to the area of ΔABC? A. ½ B. 2 C. ¼ D. 4 Which equation can be used to find the distance, d, in feet, from the basket of the hot air balloon to the base of the monument? 16

17 57. A 12-foot ladder is leaning against a building at a 75 angle with the ground. 59. In this diagram, CD is the perpendicular bisector of AB. The two-column proof shows that AC is congruent to BC. Which equation can be used to find how high the ladder reaches up the side of the building? 58. Look at quadrilateral ABCD. Which of the following would justify Step 6? A. AAS B. ASA C. SAS D. SSS Which information is needed to show that quadrilateral ABCD is a parallelogram? A. Use the distance formula to show that diagonals AC and BD have the same length. B. Use the slope formula to show that segments AB and CD are perpendicular and segments AD and BC are perpendicular. C. Use the slope formula to show that segments AB and CD have the same slope and segments AD and BC have the same slope. D. Use the distance formula to show that segments AB and AD have the same length and segments CD and BC have the same length. 60. In the triangle shown, GH ǁ DF. What is the length of GE? A. 2.0 B. 4.5 C. 7.5 D

18 61. Parallelogram FGHJ was translated 3 units down to form parallelogram F G H J. Parallelogram F G H J was then rotated 90 counterclockwise about point G to obtain parallelogram F G H J. 63. Figure A B C D F is a dilation of figure ABCDF by a scale factor of 1 2. The dilation is centered at ( 4, 1). Which statement is true about parallelogram FGHJ and parallelogram F G H J? A. The figures are both similar and congruent. B. The figures are neither similar nor congruent. C. The figures are similar but not congruent. D. The figures are congruent but not similar. 62. A regular pentagon is centered about the origin and has a vertex at (0, 4). Which statement is true? 64. A parallelogram has vertices at (0, 0), (0, 6), (4, 4), and (4, 2). Which transformation maps the pentagon to itself? A. a reflection across line m B. a reflection across the x-axis C. a clockwise rotation of 100 about the origin D. a clockwise rotation of 144 about the origin Which transformation maps the parallelogram to itself? A. a reflection across the line x = 2 B. a reflection across the line y = 2 C. a rotation of 180 about the point (2, 2) D. a rotation of 180 about the point (0, 0) 18

19 65. Which sequence of transformations maps ΔABC to ΔRST? 67. Use this triangle to answer the question. This is 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. A. Reflect ΔABC across the line x = 1. Then translate the result 1 unit down. B. Reflect ΔABC across the line x = 1. Then translate the result 5 units down. C. Translate ΔABC 6 units to the right. Then rotate the result 90 clockwise about the point (1, 1). D. Translate ΔABC 6 units to the right. Then rotate the result 90 counterclockwise about the point (1, 1). 66. Jason constructed two cylinders using solid metal washers. The cylinders have the same height, but one of the cylinders is slanted as shown. 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. Which statement is true about Jason s cylinders? A. The cylinders have different volumes because they have different radii. B. The cylinders have different volumes because they have different surface areas. C. The cylinders have the same volume because each of the washers has the same height. D. The cylinders have the same volume because they have the same crosssectional area at every plane parallel to the bases. 68. Which is an equation for the circle with a center at ( 2, 3) and a radius of 3? A. x 2 + y 2 + 4x 6y + 22 = 0 B. 2x 2 + 2y 2 + 3x 3y + 4 = 0 C. x 2 + y 2 + 4x 6y + 4 = 0 D. 3x 2 + 3y 2 + 4x 6y + 4 = 0 19

20 69. Given the points P(2, 1) and Q( 9, 6), what are the coordinates of the point on directed line segment PQ that partitions PQ in the ratio 3/2? 71. Triangle ABC has vertices as shown. What is the area of the triangle? 70. An equation of line a is y = ½x 2. Which is an equation of the line that is perpendicular to line a and passes through the point ( 4, 0)? 72. 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. If 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 license plate belongs to the set V and O? 20

21 73. For which set of probabilities would events A and B be independent? A. P(A) = 0.25; P(B) = 0.25; P(A and B) = 0.5 B. P(A) = 0.08; P(B) = 0.4; P(A and B) = 0.12 C. P(A) = 0.16; P(B) = 0.24; P(A and B) = 0.32 D. P(A) = 0.3; P(B) = 0.15; P(A and B) = Assume that the following events are independent: The probability that a high school senior will go to college is The probability that a high school senior will go to college and live on campus is 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 Given that tan(g ) =, which ratio must have a value equivalent to the tangent of g? 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 The figure shows circle C with tangent lines QR and SR. 77. Points A, B, C, D, and E are located on the circle O, as shown in this figure. 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 ½ x. C. The measure of PSR is 90. D. The measure of CQR is 90. E. The measure of QRS is (180 x). F. The measure of QRS is 2x. The measure of arc CD is 80. What is the value of x? A. 50 B. 40 C. 35 D

22 78. Look at the square WXYZ on this coordinate plane. 81. Which statement is true of the circle and the line described by the nonlinear system shown below? (x + 1) 2 + (y + 1) 2 = 25 x + y = 3 A. The line is tangent to the circle. B. The line is a secant of the circle. C. The line does not intersect the circle. D. The line passes through the center point of the circle. What is the perimeter of square WXYZ? A. 20 units B units C. 32 units D units 82. A silo used to hold grain consists of a cylinder with a height of 25 meters capped by a hemisphere with a diameter of 6 meters. 79. What is the equation of a line that is perpendicular to y = ½ x 6 and passes through the point (6, 4)? 80. 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 ¾ of his lawn only. What is the approximate volume of the silo to the nearest cubic meter? 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? 22