1 Name Geometry Exam Review #1: Constructions and Vocab Copy an angle: 1. Place your compass on A, make any arc. Label the intersections of the arc and the sides of the angle B and C. 2. Compass on A, make the same arc from #1. Label the intersection B. A 3. MEASURE from B to C. you will have to adjust your compass. 4. with your compass open from #3, compass on B, make an arc. Where this arc crosses the arc from step 2 label C. Connect A and C. A Bisect an angle: 1. Compass on A, make any arc. Label where the arc crosses the sides of the angle as B and C. 2. compass on B, make an arc 3. do not change compass: compass on C make an arc A 4. label where arcs intersect D
2 Parallel Lines: 1. Place point P above a line. Connect P to the line on some kind of angle/slant. Label the intersection of the lines Q. 2. compass on Q. Draw any arc. Label the intersection points A and B. 3. compass on P. Draw the same arc from #2. Label the intersection point C. 4. MEASURE from A to B you will have to adjust your compass. 5. with your compass open the distance from A to B, place your compass on C and make an arc, label the intersection D. 6. connect P and D. * Q and P are congruent corresponding angles
3 Perpendicular Bisector: 1. open compass to a little more than half the segment. Compass on A, make an arc. 2. do not change compass!! Compass on B, make an arc. 3. arcs must cross each other!! Label these points C and D. Connect C and D. 4. label the intersection of and E., A B *also, if you place any point along the perpendicular bisector ( ) it is equidistant to the endpoints Perpendicular from a point to a line 1. place point P above the line 2. compass on P, make an arc so it crosses the line twice. Label these A and B. 3. widen your compass a bit. Compass on A, make an arc. Without changing compass, compass on B, make an arc. 4. label where the arcs intersect as C. Connect to P.
4 Incenter: These all mean the same thing: Find the incenter Find the center of the circle that is inscribed in a triangle Construct the incircle, inscribed circle Steps: Construct each angle bisector of a triangle. The bisectors are concurrent at the incenter. Call the incenter C. Construct a line perpendicular from C to one of the sides of the triangle. This will be the radius of the circle. Compass on C, open to the length of the radius, draw a circle. It should just graze the sides of the triangle. *the incenter is equidistant to the sides of the triangle because it is the radius of the circle.
5 Circumcenter: These all mean the same thing: Find the circumcenter Find the center of the circle that you can circumscribe about a triangle Construct the circumcircle, circumscribed circle Steps: 1. construct the perpendicular bisectors of each side of the triangle. These lines are concurrent at the circumcenter. Label the circumcenter C. 2. Compass on C, open it to one of the vertices (corners) of the triangle. This is the radius of your circle. Draw a circle. *The circumcenter is equidistant to the vertices of the triangle because it is the radius of the circle. Square: 1. Draw the diameter of the circle. Label the endpoints A and B. 2. Construct the perpendicular bisector of. Make sure this line goes all the way through the circle. Label where it crosses the circle C and D. 3. Connect A, B, C, D
6 Hexagon: 1. Label the center of the circle O. Place any point A on the circle. 2. open your compass the length of. This is the radius of your circle. 3. Compass on A, make an arc crossing the circle. Label it B. 4. compass on B, make an arc. Label it C. Continue around the circle. 5. Connect A & B, B & C, and so on. Equilateral triangle: Label the center of your circle O. Place any point A on the circle. 2. open your compass the length of. This is the radius of your circle. 3. Compass on A, make an arc crossing the circle. Label it B. 4. compass on B, make an arc. Label it C. Continue around the circle. 5. Connect EVERY OTHER letter: A and C, C and E, E and A. For additional help on constructions go to mathopenref.com
7 Three undefined terms in geometry: Three transformations that use rigid motions to produce congruent figures: Congruent means: When figures are congruent, their sides are and their angles are. The transformation that produces similar figures: When figures are similar, their sides are and their angles are. Medians are concurrent at the Altitudes are concurrent at the Angle bisectors are concurrent at the Perpendicular bisectors are concurrent at the
8 Name Support Exam Review #2: Transformations Translations ( ) Notation: a translation of left 3 up 4 can also be written: Vector notation: Arrow notation/rule: (x, y) (x 3, y + 4) 1. Which is the same as a translation of right 5 and down 7? A. B. C. D. 2. Which is the same as a translation of left 6? A. B. C. D. 3. Which is the same as a translation of left 3 down 8? A. (x, y) (x + 3, y + 8) B. (x, y) (x 3, y + 8) C. (x, y) (x + 3, y 8) D. (x, y) (x 3, y 8) 4. 5.
9 Reflections ( ): Image should be equidistant from the reflection line Reflect over y = x Rotations ( ): Clockwise: counterclockwise:
10 Dilations: Sides are proportional, angles are congruent, shapes are similar What is the ratio of FG to F G? What is the ratio of F G to FG? The center of dilation: This picture is modeling that the new image is twice the distance from the origin (center (0, 0)) 15. find the scale factor: 16. Dilate the triangle with scale factor 2, center (0, 0)
11 Image Rules: V (2, -3) V R (5, -3) R Z (4, 2) Z Rule for reflection over x-axis (x, y) A (0, 2) A L (2, 0) L V (5, 4) V Rule for reflection over y-axis (x, y) T (-4, -2) T G (-2, -1) G W (-2, -4) W Rule for rotation 90 clockwise OR 270 counterclockwise (x, y)
12 E (2, 1) E N (4, -1) N F (4, 4) F Rule for rotation 180 (x, y) Rotate 270 clockwise about (0,0) T (-3, 5) T H (-3, 2) H D (-4, 3) D Rule for rotation 270 clockwise OR 90 counterclockwise (x, y) Practice: 1. What is the scale factor? What is the center of the dilation?
27 Name Support Exam Review 3 Midpoint Formula: Find the midpoint of each side of the triangle: Distance formula: Find the perimeter of the triangle: (find the length of each side, add) Find the perimeter of the triangle: (hint because it is a right triangle you can use Pythagorean theorem)
28 Find the area of the triangles: (A = ½ bh) Practice: Midpoint of RC Midpoint of CD Midpoint of RD Length RC Length CD Length RD Perimeter of triangle Area of triangle
30 Supplementary Angles Linear Pair 1 = 74 2 = 5 = (5x + 25) 7 = (5x + 15) X = 5 = 7 = Same Side Interior (Consecutive interior) 4 = = 3 = (15x- 10) 5 = (15x + 10) X = 3 = 5 = Same Side Exterior 1 = = 1 =(20x + 6) 8 = (5x + 24) X = 1 = 8 =
31 Triangle Angle Sum Theorem: the sum of the angles in a triangle = 180 (X + Y + Z = 180) Exterior Angle Theorem: The exterior angle of a triangle = sum of two remote interior angles (W = X + Y) Examples:
37 Name Support Exam Review 5 Parallel lines have the same slope and different y-intercepts Write an equation of the line parallel to the given line passing through the given point: 1. y = 2x + 4 (-3, 1) 2. y = - x + 1 (6, -2) 3. y = x 4 (-4, 6) 4. 4y = 8x + 10 (-1, 3) 5. 3x + 6y = 12 (-2, 5)
38 Perpendicular Lines have opposite reciprocal slopes Write an equation of the line perpendicular to the given line through the point 6. y = - x + 5 (2, 3) 7. y = 2x 3 (-4, 1) 8. Y = -3x + 1 (3, 6) 9. 3y + 6 = 4x (4, 8) 10. 2x 4y = -12 (-2, 5)
39 Practice: Write an equation of the line parallel to the given line through the given point: 1. y = x + 3 (5, 2) 2. Y = -3x 7 (1, 1) 3. 3y + x = 6 (3, 1) 4. 2x + y = 4 (-3, 5) 5. 4y 8 = 3x (-4, 4) 6. -4x + 2y = 10 (3, 5)
40 Write an equation of the line perpendicular to the given line through the point: 7. y = - x + 5 (-2, 6) 8. Y = 3x 2 (-6, 8) 9. 4y = 5x + 12 (-5, 2) 10. 3x 2y = 6 (6, 4) 11. y + 5 = x (-8, 2) 12. Y + 2x = 6 (2, 1)
41 Your exam has 40 questions. If you google the standard, look for the Shmoop.com entry. It will have additional practice problems (sample assignments) G.CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc G.CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
43 G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself
44 G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 14.
45 G.CO.5:Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
46 17. G.CO.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent
47 A triangle was rotated 90 degrees counterclockwise and then translated 2 down and 4 left. The final coordinates of the triangle are: (1, -3), (-2, 0), and (3, 2). What were the original coordinates? A (1, 1) (4, 4) (6, -1) B (0, 2) (2, -3) (-3, -1) C (-1, 3) (2, 0) (-3, -2) D (3, 7) (8, -2) (6, 4) G.CO.7 Use the definitions of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. 23.
48 G.CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow the definition of congruence in terms of rigid motions.
49 G.CO.9: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
51 33. G.CO.10: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. 34.
52 G.CO.11: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals
53 G.CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line
54 41. G.CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.SRT.1: Verify experimentally the properties of dilations given by a center and a scale factor:
55 44. G.SRT.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. G.SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar
56 47. G.SRT.4: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. 48.
57 G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.C.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle A (4, 6) B (6, 4) C (3, 3) D (1, 2)
58 G.GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). G.GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio Where should you plot point X so that PX is of the length of PQ? A. (4, 0) B (6, 0) C (7, 0) D (8, 0) Where should you plot X so it divides PQ in a ratio of 3:4? A. (5, 0) B (4, 0) C 4, 0) D (3, 0)
59 G.GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula
60 Additional midterm review key 1. A 2. D 3. A 4. B 5. B 6. B 7. A 8. C 9. D 10. D 11. A 12. C 13. B 14. A 15. B 16. B 17. A 18. D 19. C 20. A 21. C 22. A 23. D 24. D 25. C 26. C 27. C 28. B 29. B 30. B 31. C 32. D 33. D 34. A 35. C 36. B 37. D 38. A 39. A 40. A 41. B 42. B 43. D 44. C 45. C 46. B 47. B 48. C 49. C 50. C 51. A 52. B 53. C 54. A 55. D, A 56. B 57. D 58. C 59. B 60. C
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