Geometry Level 2 Final Exam Review Due, with work, the day of your exam!!!!!!!!

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1 Geometry Level 2 Final Exam Review Due, with work, the day of your exam!!!!!!!! In addition to reviewing all quizzes, tests, homework, and notes assigned throughout the second semester, students should always study by doing additional problems. The final exam is cumulative. The following packet consists of skill-based problems by chapter. In order to be successful on the Final Exam, students also need to know Geometry vocabulary, notation, theorems, and formulas. The following formulas will be given on the exam. Distance Formula: d = Pythagorean Theorem: a 2 + b 2 = c 2 Slope Formula: m= Area of a Triangle: A = In addition to this packet, the following pages in your book are additional problems that can be completed Chapter 6: p R6, #1-20 Chapter 8: p R8, #1-22 Chapter 7: p R7, #1-20 Chapter 10: p R10 #1-20 Unit 6- Quadrilaterals Unit 7- Similarity Sec Topic New vocab, theorems etc. 6-1 Angles of polygons Diagonal, polygon interior angles sum, polygon exterior angles sum, 6-2 Parallelograms Parallelogram, properties of parallelogram, diagonals of parallelograms, 6-3 Test for parallelograms Conditions for parallelograms, 6-4 Rectangles Rectangle, diagonals of rectangle, 6-5 Rhombi and squares Rhombus, square, diagonals of rhombus, conditions for rhombi and squares, 6-6 Trapezoid and kites Trapezoid, bases, legs of a trapezoid, base angles, isosceles trapezoid, midsegment of a trapezoid, kite, trapezoid midsegment theorem, Sec Topic New vocab, theorems etc. 7-1 Ratios and proportions Ratio, extended ratios, proportions, extremes, means, cross products, cross product property, equivalent proportions, 7-2 Similar polygons Similar polygons, scale factor, perimeters of similar polygons, 7-3 Similar triangles AA similarity, SSS similarity, SAS similarity, reflexive, symmetric, transitive properties of similarity, 7-4 Parallel lines and proportional parts 7-5 Parts of similar triangles 7-6 Similarity transformations 7-7 Scale drawings and models Midsegment of triangle, triangle midsegment theorem, proportional parts of parallel lines, congruent parts of parallel lines Special segments of similar triangles, triangle angle bisector Dilation, similarity transformation, center of dilation, scale factor of a dilation, enlargement, reduction, Scale model, scale drawing, scale, 1

2 Unit 8- Right Triangles and Trigonometry Sec Topic New vocab, theorems etc. 8-1 Geometric Mean Geometric mean, theorem 8.1 (altitude of right triangles), right triangle geometric mean theorems, 8-2 Pythagorean Theorem and Converse Pythagorean triple, converse to Pythagorean theorem, Pythagorean inequality theorems, theorem, theorem 8-3 Special right Triangle 8-4 Trigonometry Trigonometry, trimetric ratio, sine, cosine, tangent, inverse sine, inverse cosine, inverse tangent, 8-5 Angle of elevation and depression Angle of elevation, angle of depression, Unit 9-Circles (Chapter 10) Sec Topic New vocab, theorems etc Circles and Circumference Special segments in circles, circle pairs, circumference, intersecting circles 10-2 Measuring Angles and Arcs Central angles, minor and major arcs, arc addition, arc length 10-3 Arcs and Chords Congruent arcs and chords, bisecting arcs and chords, 10-4 Inscribed Angles Inscribed angles and intercepted arcs, angles of inscribed polygons 10-5 Tangents Tangent, point of tangency, common tangent, tangent perpendicular to radius, circumscribed polygons 10-6 Secants, Tangents, and Angle Measures 10-7 Special Segments in a Circle Secants to circles, circle and angle relationships formed by tangents and secants Segments in chords relationships, secant and tangent segments 10-8 Equations of Circles Equation of a circle in standard form, center, radius, completing the square? 2

3 Chapter 6: Special Quadrilaterals and Polygons Give the most specific name for the quadrilateral 1) 2) 1) 2) 3) 4) Three of the vertices of ABCD are given. Find the coordinates of point D. 5) A(3, 6), B(6, 7), C(6, 3), D(x, y) 3) 4) 5) Find the value of x 6) 6) 7) 7) 8) 8) 9) 9) 10) 10) 12) 11) 13) 11) 12) 13) 3

4 14) The measure of one interior angle of a parallelogram is 42 degrees more than twice the measure of another angle. Find the measure of each angle. The diagonals of rhombus STUV intersect at W. Given that m UVT = 31.8, TU = 20, and TW = 17, find the indicated measure. 15) m UVS 16) m TUV 17) SU 18) m TWU 19) UW 20) VT Find the sum of the measures of the interior angles of the indicated convex polygon. 21) Decagon 22) Heptagon 23) 18-gon 24) 30-gon The sum of the measures of the interior angles of a convex polygon is given. Classify the polygon by the number of sides. 25) ) ) ) 3960 Find the value of x and y in each parallelogram. 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 29) 30) 31) 32) 31) 32) 33) 34) 33) 34) 4

5 For what value of x is the quadrilateral a parallelogram? 35) 35) Find the value of n for each regular n-gon described. 36) Each interior angle of the regular n- gon has a measure of ) The measure of one interior angle of a parallelogram is 30 degrees less than 9 times the measure of another angle. Find the measure of each angle. 37) Each exterior angle of the regular n-gon has a measure of 5. 39) What is the measure of one exterior angle for a regular 40-gon Tell whether the statement is always, sometimes, or never true. 40) The diagonal of a polygon connects two adjacent vertices. 42) A hexagon is equiangular but not equilateral. 44) A hexagon has six congruent sides. 46) A triangle is concave. 48) A line containing a convex polygon s side shares interior points with the polygon. 41) A quadrilateral is convex. 43) A pentagon is a plane figure. 45) A quadrilateral is equiangular but not equilateral. 47) The exterior angle sum for a convex heptagon is ) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 5

6 Quadrilateral ABCD is a rectangle. 49) 49) If AE = 36 and CE = 2x 4, find x. 51) If BC = 24 and AD = 5y 1, find y. 53) If m AED = 12x and m BEC = 10x + 20, find m AED. 50) If BE = 6y + 2 and CE = 4y + 6, find y. 52) If m BEA = 62, find m BAC. 54) If BD = 8y 4 and AC = 7y + 3, find BD. 50) 51) 52) 53) 54) 55) 56) 55) If m DBC = 10x and m ACB = 4 6, find m ACB. 56) If AB = 6y and BC = 8y, find BD in terms of y. Quadrilateral ABCD is a rhombus. Find each value or measure. 57) 57) If m ABD = 60, find m BDC. 59) Find m CEB. 61) If AE = 3x 1 and AC = 16, find x. 63) If m CDB = 6y and m ACB = 2y + 10, find y. 58) If AB = 26 and BD = 20, find AE. 60) If m CBD = 58, find m ACB. 62) If AE = 8, find AC. 64) If AD = 2x + 4 and CD = 4x 4, find x. 58) 59) 60) 61) 62) 63) 64) 6

7 Chapter 7: Similarity List all pairs of congruent angles for the figures. Then write the ratios of the corresponding sides in a statement of proportionality. 1) ABC DFE 2) WXYZ ~ MNOP 1) 2) Triangles ABC and DEF are similar. 3) Which statement is not correct? AB BC DE EF CA AB, FD DE, 3) Determine whether the polygons are similar. If they are, write a similarity staement and find the scale factor. 4) 5) 4) In the diagram, WXYZ MNOP. 6) Find the scale factor of WXYZ to MNOP. 7) Find the values of x, y, and z. 8) Find the perimeter of WXYZ. 9) Find the perimeter of MNOP. 10) Find the ratio of the perimeter of MNOP to the perimeter of WXYZ. 5) 6) 7) 8) 9) 10) 7

8 The two triangles are similar. Find the values of the variables. 11) 12) 11) 12) Triangles RST and WXY are similar. The side lengths of RST are 10 inches, 14 inches, and 20 inches, and the length of an altitude is 6.5 inches. The shortest side of WXY is 15 inches long. Sketch each of the two triangles. 13) 13) Find the lengths of the other two sides of WXY. 15) The ratio of one side of ABC to the corresponding side of a similar DEF is 4:3. The perimeter of DEF is 24 inches. What is the perimeter of ABC? In the diagram, XYZ MNP 16) Find the scale factor of XYZ to MNP. 17) Find the unknown side lengths of both triangles. 18) Find the length of the altitude shown in XYZ. 19) Find and compare the areas of both triangles. 14) Find the length of the corresponding altitude in WXY. 18 inches 24 inches 32 inches 14) 15) 16) 17) 18) 19) The community park has a rectangular swimming pool enclosed by a rectangular fence for sunbathing. The shape of the pool is similar to the shape of the fence. The pool is 30 feet wide. The fence is 50 feet wide and 100 feet long. 20) 20) What is the scale factor of the pool to the fence? 22) What is the length of the pool? 21) Find the area reserved strictly for sunbathing. 21) 22) 23) 8

9 Use the diagram to complete the statements. 24) 23) ABC ~? 25) AB? CA 26) 24)? EF? 25) B? 27)? 8 28) 26) 12? 27) x =? 28) y =? 29) Determine whether the triangles are similar. If they are, write a similarity statement. 30) 31) 29) 30) 32) 33) 31) 32) 34) 33) 34) 9

10 35) In the diagram at the right, find the length of BC 35) 36) List three pairs of congruent angles. 37) Name two pairs of similar triangles and write a similarity statement for each. 38) Is ACD ~ BCE? 39) ls AED EAB? 40) Triangles ABC and DEF are right triangles that are similar. AB and BC are the legs of the first triangle. DE and EF are the legs of the second triangle. Which of the following is false? A. A D B. AC = DF AC AB C. = DF DE 36) 37) 38) 39) 40) In order to estimate the height h of a flag pole, a 5 foot tall male student stands so that the tip of his shadow coincides with the tip of the flag pole s shadow. This scenario results in two similar triangles as shown in the diagram. 41) Why are the two overlapping triangles similar? 41) 42) 43) 42) Using the similar triangles, write a proportion that models the situation. 43) What is the height h (in feet) of the flag pole? 44) Is either LMN or RST similar to ABC? 45) Is either LMN or RST similar to ABC? 44) 45) 10

11 Show that the triangles are similar and write a similarity statement. Explain your reasoning. 46) 47) 46) 48) In the diagram at the right, ACE ~ DCB. Find the length of AB. 47) 48) Use the diagram at the right to copy and complete the statement 49) ABC ~? 50) m DCE =? 51) AB =? 52) m CAB + m ABC =?_ 49) 50) 51) 52) In order to estimate the height h of a tall pine tree, a student places a mirror on the ground and stands where she can see the top of the tree, as shown. The student is 6 feet tall and stands 3 feet from the mirror which is 11 feet from the base of the tree. 53) What is the height h (in feet) of the pine tree? 54) Another student also wants to see the top of the tree. The other student is 5.5 feet tall. If the mirror is to remain 3 feet from the student's feet, how far from the base of the tree should the mirror be placed? 53) 54) 11

12 Use the figure to complete the proportion 55) GC? 56) 55) CF DB AF? 57) 56) FC BD CD GD 58) 57) FB? AE GE 59) 58) CD? GD? 60) 59) GE AE FG FB 60) AG? Use the given information to determine whether BD AE. 61) 62) 61) 62) 63) 64) 63) 64) Determine the length of each segment 65) GB 66) FC 67) CD 68) BC 65) 66) 67) 68) 12

13 find the value of x. 69) 70) 71) 69) 70) 72) 73) 71) 74) Find the value of the variable. 72) 75) 73) x 74) m 75) a BC DE Determine whether the given information implies. Explain. 76) 76) 77) 77) 78) 78) 79) 79) 13

14 Find the value of the variable. 80) 81) 80) 81) 82) 82) 14

15 Chapter 8: Right Triangles 1) Which equation is not correct? A 2 t r 2 = s 2 B 2 t + r 2 = s 2 C s 2 2 t = r 2 D 2 t s 2 = r 2 1) 2) A 25-foot ladder leans against a wall 7 feet from the base of the wall. How high up the wall does the ladder touch? 3) Find the area of the rectangle. 2) 3) 4) Classify ABC if the vertices are A( 2, 5), B(12, 5), and C(10, 17). 5) Find the approximate area of the triangle. 6) Find sin F and sin G. 4) 5) 6) 7) Which equation could be used to find the value of x in the diagram? 12 x A cos 55 = x B cos 35 = x C cos 35 = x D cos 55 = 12 7) 8) Which is not enough given information needed to solve a right triangle? A two acute angles and one side length B measure of the hypotenuse C two side lengths D one side length and the measure of one acute angle 8) 9) Find m A. 9) 15

16 10) What is the value of x? Round your answer to the nearest tenth. 10) 11) What is the value of x? Round your answer to the nearest tenth. 11) 12) What is the value of y? Round your answer to the nearest tenth. 12) 13) A shipping dock has a mobile ramp that is used to help load and unload cargo from trucks. The ramp is 125 inches long and has a base that is 120 inches long. What is the height h of the ramp? Real Estate An investor owns a triangular plot of land as shown in the diagram. 14) Find the perimeter of the plot of land. 15) Find the area of the triangular plot. 13) 14) 15) 16) 16) One acre of land is equivalent to 43,560 square feet. 17) How many acres are in this plot of land? Round to two decimal places. The investor is planning on selling the land. The market rate in this area is $5000 per acre. How much should the investor ask for the land? 17) 16

17 Garden You have a garden that is in the shape of a right triangle with the dimensions shown. 18) Find the perimeter of the garden. 18) 19) You are going to plant a post every 15 inches around the garden's perimeter. How many posts do you need? 19) 20) 20) If each post costs $1.25 and each foot of fencing costs $.70, how much will it cost to enclose the garden? Explain. Maps The distances between three towns are given in the diagram. 21) Is the triangle ( ABC) formed by the three towns a right triangle? 21) 22) Town B is directly west of town C. Is town A directly north of town C? 22) 23) Complete the table. Give exact answers. 23) table X Y

18 24) Complete the table. Give exact answers. 24) table _ a 9 11 b c 16 You are replacing the roof on the house shown, and you want to know the total area of the roof. The roof has a 1-1 pitch on both sides, which means that it slopes upward at a rate of 1 vertical unit for each 1 horizontal unit. 25) 25) Find the values of x and y in the diagram. 26) Find the total area of the roof to the nearest square foot. Find the value of each variable. Round decimals to the nearest tenth. 27) 28) 26) 27) 28) 29) 29) 30) 30) 31) 32) 31) 32) 18

19 33) Highway You are traveling along a stretch of highway that has a slight grade with an angle of inclination of 5. After traveling for 4 miles, what is the vertical v and horizontal h change in feet? (1 mi = 5280 ft) Round your answer to the nearest foot. 34) Skyscraper You are a block away from a skyscraper that is 780 feet tall. Your friend is between the skyscraper and yourself. The angle of elevation from your position to the top of the skyscraper is 42. The angle of elevation from your friend s position to the top of the skyscraper is 71. To the nearest foot, how far are you from your friend? 33) 34) 35) 35) Ladder You lean a 16 foot ladder against the wall. If the ladder makes an angle of 70 with the ground, how far away from the wall is the base of the ladder? Round your answer to the nearest tenth of a foot 36) Skyscraper You are standing 350 feet away from a skyscraper that is 750 feet tall. What is the angle of elevation from you to the top of the building? 36) 37) 19

20 Chapter 10: Circles For Exercises 1-7, refer to 1. Name the circle. 2. Name radii of the circle. 3. Name chords of the circle. 4. Name diameters of the circle. 5. If AB = 18 millimeters, find AR. 6. If RY = 10 inches, find AR and AB. 7. Is? Explain. Find the diameter and radius of a circle with the given circumference. Round to the nearest hundredth. 8. C = 40 in. 9. C = 256 ft 10. C = m 11. C = 9 cm 12. C = 79.5 yd Find the exact circumference of each circle using the given inscribed or circumscribed polygon Find the value of x

21 and are diameters of O. Identify each arc as a major arc, minor arc, or semicircle of the circle. Then find its measure. 20. m 21. m 22. m 23. m Use O to find the length of each arc. Round to the nearest hundredth. 24. if the radius is 2 meters 25. if the diameter is 7 inches 26. if BE = 24 feet 27. if DO = 3 millimeters Use P to find the length of each arc. Round to the nearest hundredth. 28., if MP = 2 inches 29., if PS = 10 centimeters ALGEBRA Find the value of x in each circle M P 21

22 In P, the radius is 13 and RS = 24. Find each measure. Round to the nearest hundredth. 38. RT 39. PT 40. TQ In A, the diameter is 12, CD = 8, and m = 90. Find each measure. Round to the nearest hundredth. 41. m 42. FD 43. AF 44. In R, TS = 21 and 45. In Q,, GQ = x + 5 and UV = 3x. What is x? EQ = 3x 6. What is x? Find each measure. 46. m 47. m N 48. m ALGEBRA Find each measure. 49. m U 50. m A ALGEBRA Find each measure. 51. x 52. x 22

23 ALGEBRA Find each measure. 53. m R 54. m W Find x. Assume that segments that appear to be tangent are tangent For each figure, find x. Then find the perimeter Find each measure. Assume that segments that appear to be tangent are tangent. 62. m m m 23

24 65. m 5 Find each measure. Assume that segments that appear to be tangent are tangent. 66. m 67. m m V 69. m Find x. Assume that segments that appear to be tangent are tangent. Round to the nearest tenth if necessary

25 Find x. Round to the nearest tenth. Assume segments that appear to be tangent are tangent Write the equation of each circle. 80. center at (2, 4), radius center at ( 1, 4), radius 2 Write the equation of each circle. 82. center at ( 2, 6), diameter center at origin, diameter center at (3, 4), passes through ( 1, 4) 85. center at (0, 3), passes through (2, 0) For each circle with the given equation, state the coordinates of the center and the measure of the radius. Then graph the equation = = 9 25