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GEOMETRY A Semester Exam Review Geometry A Semester Exam Review © MCPS GEOMETRY A Semester Exam Review Right Triangles Pythagorean Theorem In right triangle ABC with right angle at point C: 2 2 a b c 2 A c b B C a Trigonometry In a right triangle with acute angle A: sin A side opposite A hypotenuse cos A side adjacent to A hypotenuse tan A side opposite A side adjacent to A © MCPS GEOMETRY A Semester Exam Review Items on this review are grouped by Unit/Topic Unit 1, Topic 1 1. Write the three undefined terms in geometry. 2. Construct the angle bisector of angle A below. A 3. Construct a 45o angle using point B as the vertex of the angle. B © MCPS Review Page 1 GEOMETRY A 4. Semester Exam Review A road construction crew wishes to construct a new road parallel to the current road, and passing through the town of Herkimer, marked by point H below. Construct the location of the road current road H 5. The perpendicular bisector of AB is constructed. What is true about every point on that perpendicular bisector? 6. The angle bisector of CDE is constructed. What is true about every point on the angle bisector? 7. A line is constructed parallel to a given line. What is true about the lines? © MCPS Review Page 2 GEOMETRY A Semester Exam Review Unit 1, Topic 2 8. Point A 1, 4 is to be transformed to point A using the translation rule x, y x 3, y 6 . What are the coordinates of point 9. A ? Triangle ABC has been transformed to triangle ABC . y A C A B O x C B a. State in words, the transformation(s) that produce triangle ABC . b. Write a function rule that represents this transformation. c. Why must ABC ABC ? d. On the coordinate plane above, sketch the reflection of ABC across the x-axis. Label the triangle DEF. Write the function rule for this transformation. © MCPS Review Page 3 GEOMETRY A 10. Semester Exam Review Let 3,5 be a point on the coordinate plane as shown below. y 3,5 O x Write the coordinates of the image point if 3,5 undergoes the following transformations. © MCPS a. Reflected across the x-axis. b. Reflected across the y-axis. c, Translated seven units to the right and two units downward. d. Rotated clockwise 90 degrees about the origin. e. Rotated 180 degrees about the origin. f. Rotated counter-clockwise 90 degrees about the origin. Review Page 4 GEOMETRY A Semester Exam Review 11. y O A D B x C a. Reflect the figure above across the line y 1 . Name the image ABC D . b. Does this transformation preserve lengths and angle measurements? Justify your answer. c. If the figure was reflected across the x-axis, then translated two units upward, would the result be the same as the transformation in part a) ? © MCPS Review Page 5 GEOMETRY A Semester Exam Review 12. Write the three rigid transformations. 13. If a figure undergoes a rigid transformation, then the transformed figure must be ___________________ to the original figure. 14. Look at the figure below. y O x Complete each statement such that each transformation will map the triangle onto itself. a. A reflection across the _____ axis or the line x _____ . b. A rotation of _______ degrees about the point _________. © MCPS Review Page 6 GEOMETRY A 15. Semester Exam Review Triangle ABC undergoes a transformation to produce triangle ABC . The triangles are shown below. y A C B O x B C A a. Is the transformation above a single reflection? Justify your answer. b. Is the transformation above a translation? Justify your answer. c. Determine a transformation with three reflections that will produce triangle ABC . © MCPS Review Page 7 GEOMETRY A Semester Exam Review Unit 1, Topic 3 16. If figure ABCD is congruent to figure EFGH, write all eight side and angle congruence statements. 17. If all corresponding sides of two figures are congruent, and all corresponding angles of the two figures are congruent, then the figures must be ________________. In the figure below, point S is the midpoint of WU . Therefore WS US . R W T S U You wish to prove WSR UST . 18. To prove this congruence by SSS, what two additional congruence statements are needed? 19. To prove this congruence by SAS, what two additional congruence statements are needed? 20. To prove this congruence by ASA, what two additional congruence statements are needed? 21. To prove this congruence by AAS, what two additional congruence statements are needed? 22. State two additional congruence statements that will be insufficient to prove the triangles congruent. © MCPS Review Page 8 GEOMETRY A 23. Semester Exam Review Consuela wants to determine the length of a power line that will be stretched over a lake. She cannot walk through the lake. She was able to take some measurements, hoping to determine the length. Her measurements are shown below. Figure NOT drawn to scale 625 ft 450 ft 500 ft 500 ft 450 ft Power Line Consuela believes that the length of the power line is 625 feet, but she’s not sure how to explain this to her boss. Using what you know about triangle congruence, help Consuela by writing a brief report as to why the length of the power line is 625 feet. © MCPS Review Page 9 GEOMETRY A Semester Exam Review Look at triangles ABC and FED below. E B A C D F In items 24 through 29 below, information is given about the two triangles. State whether the triangles can be proven congruent by ASA, SSS, AAS, or SAS. If the triangles cannot be proven congruent, state why. 24. AB FE , AC FD, A F . 25. A F , B E , C D . 26. A F , B E , AB FE . 27. AB FE , AC FD, BC ED . 28. B E , AC FD, BC ED . 29. A F , AB FE , C D . © MCPS Review Page 10 GEOMETRY A Semester Exam Review Unit 1, Topic 4 30. Every quadrilateral that is a parallelogram has certain properties. List all these properties. 31. What properties does a rectangle have in addition to those of a parallelogram? 32. What properties does a square have in addition to those of a rectangle? 33. Triangle DEF is isosceles with D E . Which sides are congruent? 34. In the figure to the right, D is the midpoint of AB and E is the midpoint of AC . Figure NOT drawn to scale Determine the values of x, y, and z. 22 C B 10 x y D E z 12 A © MCPS Review Page 11 GEOMETRY A 35. Semester Exam Review In the figure below, D is the midpoint of AB and E is the midpoint of AC . C B D E A Complete the statements using the items in the box on the right. Each item may be used more than once or not at all. a. b. AD _______ AE _______ c. _______ is parallel to _________ d. AD _______ e. AE _______ f. _______ = 2 _______ g. The ratio AB : AD is ________ h. The ratio AE : AC is ________ © MCPS AD AE DE BC EC BD DE AD BC EC BD AE 2 :1 1: 2 Review Page 12 GEOMETRY A 36. Semester Exam Review If two parallel lines are cut by a transversal: a. Which angle pairs are congruent? b. Which angle pairs are supplementary? p 37. Given: m n, 1 16 1 2 3 4 Prove: p q 9 10 11 12 38. Given: © MCPS 7 5 6 8 m 13 14 15 n 16 D A A D C AB DE Prove: CE CB q B E Review Page 13 GEOMETRY A Semester Exam Review D A C is the midpoint of BE. 39. Given: AB DC AB BE , DC BE Prove: A D B © MCPS C E Review Page 14 GEOMETRY A Unit 2, Topic 1 Semester Exam Review Figure NOT drawn to scale 30 C B 40. In the figure to the right, DE BC . a. Prove that ADE ~ ABC . 15 x y D E 9 12 A b. Determine the value of x. c. Determine the value of y. © MCPS Review Page 15 GEOMETRY A 41. Semester Exam Review Jory wants to measure the length, L, of a pond. The figure below shows the measurements she will use to determine the length of the pond. In the figure below, AE and BD intersect at point C. 30 ft A B 25 ft 20 ft C 40 ft 50 ft D Figure NOT drawn to scale L E ABC EDC ? a. Which similarity postulate/theorem may be used to prove b. What is the length, L, of the pond? Show how you determined your answer. © MCPS Review Page 16 GEOMETRY A 42. 43. Semester Exam Review If two figures are similar, then a. Corresponding sides are _________________________(proportional/congruent). b. Corresponding angles are ________________________(proportional/congruent). In the figure below, QRS ~ QTU . S Q U R T a. Prove that RS TU . b. Write all congruences and proportions using the sides and angles in the figure. © MCPS Review Page 17 GEOMETRY A 44. Semester Exam Review The figure below shows BCD . BC D will be the image of BCD after a dilation with center B and scale factor 5. Item Bank B C B C D Complete the statements using the item bank on the right. D 5 1 5 parallel to a. The point B will be the same point as point _____. on the same line as b. C D will be _______________ CD . perpendicular c. BC will be _______________ BC . d. The perimeter of BC D will be _____ times the perimeter of BCD . © MCPS Review Page 18 GEOMETRY A 45. Semester Exam Review On the graph below, quadrilateral ABCD has been dilated, with the center of dilation the origin, to create quadrilateral ABC D . y 10 B 9 8 7 6 B 5 A 4 3 A 2 1 -10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1O D D -1 1 -2 -3 -4 2 3 4 5 6 7 8 C 9 10 x C -5 -6 -7 -8 -9 -10 a. What is the scale factor of the dilation? b. What is the ratio of the length of AB to the length of AB ? c. How are the measures of A and A related? © MCPS Review Page 19 GEOMETRY A 46. Semester Exam Review Triangle ABC is shown on the coordinate grid below. y 10 9 8 7 6 5 4 3 2 1 -10 - 9 - 8 - 7 - 6 - 5 - 4 - 3 - 2 - 1 O C A 1 2 3 4 5 6 7 8 9 10 x -1 -2 -3 -4 B P -5 -6 -7 -8 -9 -10 Triangle ABC is to be dilated by a scale factor of three with the center of dilation the point P 5, 6 to produce triangle ABC . a. On the coordinate plane above, sketch ABC . b. What is the ratio of the perimeter of ABC to the perimeter of ABC ? c. What is the relationship between A and A ? © MCPS Review Page 20 GEOMETRY A 47. Semester Exam Review Pentagons ABCDE and FGHIJ are similar. B A Figure NOT drawn to scale F 10 C J G 15 E D I 12 x H What is the value of x? 48. Derrick is creating a model of the Eiffel Tower. The Eiffel Tower is 301 meters tall and has a base of width 100 meters. His model will have a base of 10 cm. What will be the height of the model? © MCPS Review Page 21 GEOMETRY A 49. Semester Exam Review Jack wishes to find the height of a tree. He puts a stick into the ground and uses the sun’s shadow to take some measurements. The figure below shows his measurements. Figure NOT drawn to scale h ft 5 ft 135 ft 9 ft a. Determine the height of the tree. b. Determine the distance from the tip of the sun’s shadow to the top of the tree. © MCPS Review Page 22 GEOMETRY A Semester Exam Review 50. a. Draw examples of two triangles that are similar by SSS Similarity. b. Draw examples of two triangles that are similar by AA Similarity. c. Draw examples of two triangles that are similar by SAS Similarity. © MCPS Review Page 23 GEOMETRY A Semester Exam Review Unit 2, Topic 2 51. Given right ABC , use the item bank to identify the following. Items may be used more than once. A B C Item Bank A B C BC AC BC AB AB BC AB AC AB AC AC BC BC AC AB a. Leg opposite A h. Tangent ratio of C b. Leg opposite C i. The angle whose Sine ratio is c. Sine ratio of A j. The angle whose Tangent Ratio is d. Cosine ratio of A k. Hypotenuse e. Tangent ratio of A f. Sine ratio of C g. Cosine ratio of C © MCPS BC AC AB BC Review Page 24 GEOMETRY A 52. Semester Exam Review Let ABC be a right triangle, with right angle at C. Which statements are true? Check all that apply. A _____ A and B are complementary. _____ m A 90o m B B sin A _____ tan A cos A 2 2 _____ AB AC BC C 2 _____ sin A cos B _____ cos A sin 90o A _____ If A B, then sin A sin B 53. Look at the two triangles below. 16 20 12 12 20 A 16 B Saundra says that since the two triangles are congruent that sin A sin B . Is Saundra correct? Justify your answer. © MCPS Review Page 25 GEOMETRY A 54. Semester Exam Review Look at the figure below. Figure NOT drawn to scale Q 12 R 8 S 6 10 15 T 15 U a. Show that the value of cos S is the same if SQU or SRT is used. b. What is the value of sinU ? c. Is cos U cos T ? Why or why not? © MCPS Review Page 26 GEOMETRY A 55. Semester Exam Review A skateboarder has a piece of plywood that she wants to use as a ramp. The plywood is 16 feet long. She wants to put a vertical support under the ramp so that the ramp is at a 20o angle to the ground. This is shown in the figure below. The view is from the side of the ramp. 16 ft. ramp Vertical support Figure NOT drawn to scale 20o How high is the vertical support? Round your answer to the nearest hundredth of a foot. 56. Josh has a 17 ft. ladder. He would like to place the ladder against the side of a vertical wall so that the top of the ladder is 15 feet up the wall. Safety guidelines state that the angle that the ladder makes with the horizontal can be no more than 70 degrees. a. Can Josh safely place the ladder against the wall? b. Using the safety guidelines, what is the highest distance that the top of the 17-foot ladder can be above the ground? Round your answer to the nearest hundredth of a foot. © MCPS Review Page 27 GEOMETRY A 57. 58. Semester Exam Review A plane takes off from a runway and climbs at an angle of elevation of 15o. a. After the plane has travelled one mile (5280 feet) how far has the plane travelled horizontally? Round your answer to the nearest foot. b. After the plane has travelled one mile (5280 feet) how far has the plane travelled vertically? Round your answer to the nearest foot. A farmer wishes to make separate sections of a piece of land as shown in the figure below. All triangles are right triangles. 65 m Figure NOT drawn to scale 20 m x y 15 m a. What is the value of x? _______ b. What is the value of y? _______ © MCPS Review Page 28 GEOMETRY A 59. Semester Exam Review A swimmer tries to swim straight across a stream. The stream is 100 feet wide. A current in the stream, perpendicular to his intended path, caused him to land 20 feet downstream from his intended landing point. Figure NOT drawn to scale 20 ft. Intended path Actual path 100 ft a. How far did the swimmer actually swim? Give your answer to the nearest tenth of a foot. b. At what angle, , was his actual path to the intended path? Give your answer to the nearest tenth of a degree. © MCPS Review Page 29