Geometry First Semester Practice Final (cont)

49. Determine the width of the river, AE, if A. 6.6 yards. 10 yards C. 12.8 yards D. 15 yards Geometry First Semester Practice Final (cont) 50. In the similar triangles shown below, what is the value of x? A. 13.5. 16.7 C. 18 D. 24 18 J K 15 L 20 x M 51.. Determine the length of the longest side of. A. 222. 576 A C. 648 D. 666 192 222 C 216 Z N 576 X Y 52. Which additional piece of information would prove that? A. NM = 18 N K. LM = 18 C. NM = 15 12 8 12 D. LM = 10 L M I 10 J 53. For,. Find the length of A. 3. 12 C. 10 D. 20 X (cont) Q 6 Y R 14 7 Z Geometry 101

Geometry First Semester Practice Final (cont) 54. Lines l, m, and n are parallel. Determine the value of x. A. 2.7. 6.0 C. 8.5 D. 9.4 55.. Determine the value of x. A. 35. 5 C. 20 D. 30 20 56. Determine the length of. A. 19. 39 C. 52 D. 14 16 8 A D 12 x C 4 N 15 M 6 (2x + 1) 5 7. 5 x 25 l m n L A (3x 5) C Geometry 102

Unit 7 Objective 0 Determine the value of the missing angle in each of the figures below: NOTE: DIAGRAMS ARE NOT DRAWN TO SCALE. 1. 2. 3. 4. 5. 6. 7. 8. (cont) Geometry 103

Unit 7 Objective 0 (cont) Determine the value of the missing angle in each of the figures below: NOTE: DIAGRAMS ARE NOT DRAWN TO SCALE. 9. 10. 11. 12. 13. 14. Geometry 104

Geometry Unit 7 1. Apply the Pythagorean Theorem (Section 7.1) 2. Use the converse of the Pythagorean Theorem (Section 7.2) 3. Use Similar Right Triangles (Section 7.3) 4. Special Right Triangles (Section 7.4) 5. Apply the Tangent Ratio (Section 7.5) 6. Apply the Sine and Cosine Ratios (Section 7.6) 7. Solve Right Triangles (Section 7.7) Review Geometry 105

Unit 7 Trig Worksheet 1 Remember SOH CAH TOA sin = cos = tan = C Example 1: Find the value of cos A tan 5 4 Solution: cos A tan C = 3 = = Example 2: Find the value of sin 2 A 12 13 Solution: sin 2 A = (sin A) (sin A) = C 5 A Homework = = Use shown to find the values in problems 1-6 10 N 6 L 8 1. sin L cos L 2. tan N sin L 3. cos L 2 tan N M 4. tan 2 L 5. 1 + cos 2 L 6. sin 2 L + cos 2 L (cont) Geometry 106

Unit 7 Trig Worksheet 1 (cont) Use the triangle shown to find the values in problems 7-15. 7. sin A cos 8. tan sin A 3 5 9. sin 2 A 1 10. 1 cos 2 A 11. tan 2 + 1 12. C 4 A 13. 14. 15. Geometry 107

Notes: Unit 7 Trig Worksheet 2 Let s review the formulas for sin, cos, tan. sin = cos = tan = We have been finding these values using triangles with numbers. C Example 1 Find sin A 5 4 Solution: sin A = = A 3 2 In this lesson we are not going to have numbers on the triangle. Example 2 Find sin K and tan R R P K Solution: sin K = (just put the letters of the sides) = tan R = = Example 3 Determine if the statement is true or false using the given triangle. C A Solution: A cos = C (get the trig word by itself by dividing both sides by A) cos = cos = = which is the cos formula, so True (cont) Geometry 108

Unit 7 Trig Worksheet 2 (cont) Homework: Using the given triangle find the requested values in problems 1-4 (Just put the letters of the sides.) U V W 1. tan W 2. sin U 3. cos U 4. tan U Draw your own right triangle and label it JKL. Use this triangle to find the values in problems 5-7. (Just put the letters of the sides.) is a right angle. 5. sin J 6. tan L 7. cos J Use the given triangle to determine whether the statements in problems 8-15 are true or false. 8. sin A = 9. cos = 10. tan A = 11. cos A = (In Problems 12-13 remember to get the trig word by itself by dividing.) 12. A sin A = C 13. C tan = AC C A (In problems 14-15, put all the letters of the sides and compare the left side of the equation with the right side of the equation.) 14. sin A = cos 15. sin A = tan A cos A Geometry 109

Unit 7 Trig Worksheet 3 Notes: Determine whether each equation below is true or false: 1. cos 2 A + 1 = sin 2 A 2. sin 2 A + cos 2 A = 1 Solution: a) First make a right triangle. b) Put any right triangle Pythagorean numbers on your triangle. You could use 3,4,5 or 5,12,13. Let s use 3,4,5 (Remember the largest number is the hypotenuse. Also label one of the non-right angles as A c) Now, find sin A = Find cos A = 4 5 3 A d) Now plug the fractions into each problem and see if it s true or false. 1. cos 2 A + 1 = sin 2 A 2. sin 2 A + cos 2 A = 1 + 1 = + = 1 + 1 = + = 1 = = 1 This is not a true equation, so the original trig problem was false. This is a true statements, so the original trig problem was true. (cont) Geometry 110

Unit 7 Trig Worksheet 3 (cont) Homework: Use the same triangle that we used in our note examples and give the following fraction values. 1. sin A = 2. cos A = 3. tan A = Now, using the above fraction values, determine whether each of the following is true or false. 4. sin A cos A = tan A 5. tan A cos A = sin A 6. tan A sin A = cos A 7. 1 sin 2 A = cos 2 A 8. cos 2 A sin 2 A = 1 9. cos A = 10. 1 + tan 2 A = cos 2 A 11. tan A = 12. sin 2 A = tan 2 A cos 2 A Unit 7 Review NOTE: Diagrams are not drawn to scale. Select the correct multiple choice response: 1. The lengths of the two legs of a right triangle are 4 and 7. What is the length of the hypotenuse? A. 33. C. 65 D. 2. The lengths of the two legs of a right triangle are and 5. What is the length of the hypotenuse? A.. C. D. (cont) Geometry 111

Unit 7 Review (cont) 3. The length of one leg of a right triangle is and the hypotenuse is 11. Find the length of the other leg. A. 25. 5 C. D. 4. A baseball diamond is in the shape of a square, 90 feet on each side. What is the direct distance from home plate to second base? A. 90 ft. ft C. ft D. 180 ft 5. A model rocket is launched. It rises to a point 36 feet above the ground and is 48 feet along the ground from the lift off site, as shown in the diagram. What is the length of the rocket s path in the air? A. 12 ft.. 32 ft. C. 60 ft. x D. 84 ft. Current Site of Rocket 36 ft Rocket Launch Site 48 ft 6. Which set of numbers can represent the side lengths of an obtuse triangle? A. 5, 10, 11. 3, 5, C. 3, 7, 8 D. 1, 2, 2 7. Which set of numbers below represent the lengths of the sides of a right triangle? A. 1, 2,. 5, 11, 12 C. 6, 8, D. 5, 7, 9 (cont) Geometry 112

Unit 7 Review (cont) A 8. Determine the length of D. A. 13. C. 36 D. 6 9 D 4 C 9. Determine the value of x. A.. x C. D. 6 1 10. Determine the value of x. A.. C. 13 D. x 12 5 11. Determine the value of x A.. 6 C. 3 D. x 45 12. Determine the height of the triangle if A = 16 cm and C = 17 cm A. cm C. cm C. 9 cm D. 15 cm A h 13. Determine the values of x and y. A. x =, y = 3. x =, y = C. x = 3, y = D. x =, y = x 60 6 y (cont) Geometry 113

14. Determine the value of x. Unit 7 Review (cont) A.. C. D. 40 x 60 15. Which equation is equivalent to cos A =? A. x =. x = C. x = 17 cos A D. x = cos (17A) 16. Determine the value of x to the nearest tenth. A. 3.3. 2.7 C. 7.1 D. 6.4 17. In right triangle DEF, DE = 15, EF = 36 and DF = 39. What is the cos F? A. 3 25 x. C. D. 18. An 80 foot support wire is attached to the top of a tower and meets the ground at a 70 angle. How tall is the tower, to the nearest foot? A. 27 ft.. 70 ft. C. 220 ft. D. 75 ft. sin 70 = 0.94 sin 20 = 0.34 cos 70 = 0.34 cos 20 = 0.94 tan 70 = 2.75 tan 20 = 0.36 (cont) Geometry 114

Unit 7 Review (cont) 19. A ladder is leaning against a tree and hits the tree at a point 15 feet above the ground. The ladder and the ground form a 62 angle. How far, to the nearest tenth of a foot, is the bottom of the ladder from the base of the tree? A. 13.2 ft.. 7 ft. C. 28.2 ft. D. 8 ft. sin 62 = 0.88 sin 28 = 0.47 cos 62 = 0.47 cos 28 = 0.88 tan 62 = 1.88 tan 28 = 0.53 20. Which expression is correct? A. cos =. cos = f d C. cos = D. cos = P e 21. The angle of depression from the top of a 20-foot lighthouse to a boat in the ocean is 39. Which is closest to the distance that the boat is from the base of the lighthouse? A. 12.6 ft. 16.2 ft C. 24.6 ft D. 15.6 ft sin 39 = 0.63 sin 51 = 0.78 cos 39 = 0.78 cos 51 = 0.63 tan 39 = 0.81 tan 51 = 1.23 22. The angle of elevation from the boy to the top of the flagpole is 35. How far (to the nearest tenth) is the boy from the base of the flagpole? A. 85.7 ft.. 42 ft. C. 49 ft. D. 34.4 ft sin 35 = 0.57 sin 55 = 0.82 cos 35 = 0.82 cos 55 = 0.57 tan 35 = 0.70 tan 55 = 1.43 Geometry 115

Unit 8 Objective 0 1. If possible, draw 2 obtuse triangles that are similar. Label their measurements and state whether they are similar by AA, SSS, or SAS. If not possible, state that it can t be done. 2. If possible, draw 2 obtuse triangles that are not similar. Label their measurements and state why they are not similar. 3. If possible, draw 2 scalene triangles with congruent bases that are similar. Label their measurements and state whether they are similar by AA, SSS, or SAS. If not possible, state that it can t be done. 4. If possible, draw 2 scalene triangles with congruent bases that are not similar. Label their measurements and state why they are not similar. 5. If possible, draw 2 right triangles that are similar. Label their measurements and state whether they are similar by AA, SSS, or SAS. If not possible, state that it can t be done. 6. If possible, draw 2 right triangles that are not similar. Label their measurements and state why they are not similar. 7. If possible, draw 2 isosceles triangles with congruent vertex angles that are similar. Label their measurements and state whether they are similar by AA, SSS, or SAS. If not possible, state that it can t be done. 8. If possible, draw isosceles triangles with congruent vertex angles that are not similar. Label their measurements and state why they are not similar. 9. In the figure below, determine if the value of n could be 1. Explain why or why not? n n 8 10. In the figure below, determine if the value of n could be 4. Explain why or why not? n n 8 11. In the figure below, determine if the value of n could be 5. Explain why or why not? n n 12. The complement of an angle is four times the measure of the angle itself. Calculate the angle and its complement. (cont) 8 Geometry 116

Unit 8 Objective 0 (cont) 13. The measures of the interior angles of a triangle are (6x + 6), (4x 4), and (4x + 10). Calculate the degree measures of all three angles. 14. The four sides of the figure will be folded up and taped to make an open box. What will be the volume of the box? 15. Calculate the value of x. 16. Calculate the value of x. Y 17. Supply the reasons in the proof below: Given: ; bisects Prove: Statements 1. ; bisects 1. 2. 2. 3. 3. 4. 4. X Reasons W Z Geometry 117

Geometry Unit 8 1. Find angle measure in polygons. Interior angle sums, exterior angle sums, etc. (Section 8.1) 2. Use properties of parallelograms. (Section 8.2) 3. Show that a quadrilateral is a parallelogram. (Section 8.3) 4. Properties of rhombuses, rectangles, and squares (Section 8.4) 5. Use properties of trapezoids and kites. (Section 8.5) 6. Identify special quadrilaterals (Section 8.6) Review Geometry 118

Unit 8 Worksheet 2 Determine the values for the variables that make the quadrilateral a parallelogram. 1. 2. 3. 11 m + 2n 2x + 4y x + 3y 16 3m + n 7 15 a b 5 a + b 9 Determine the coordinates of the point where the diagonals of the parallelogram intersect. 4. 5. y y (5,7) (12,7) (m, n) (m + w, n) (3,4) (10,4) x x (0, 0) (w, 0) Worksheet 3 In parallelogram ACD, diagonals and intersect at point V. y theorem 8.10 we know that diagonals bisect each other so V = DV and AV = CV as shown. We also know that opposite sides are congruent in a parallelogram. Use this information and the diagram to answer true or false to the statements below: 1. 2. 3. A 4. V D C Geometry 119

Unit 8 Practicing Proofs 1. Given: Parallelogram PQRS; S K R Prove: P J Q Statements Reasons 1. PQRS; 1. 2. 2. 3. 3. 4. 4. 5. 5. 2. Given: ; Prove: QRST is a parallelogram T 3 2 S Q 1 4 R Statements Reasons 1. ; 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. QRST is a parallelogram. 6. (cont) Geometry 120

Unit 8 Practicing Proofs (cont) 3. Given: D y x C Prove: is a parallelogram. Statements A x y Reasons 1. ; 1. 2. 2. 3. 3. 4. 4. 5. is a parallelogram 5.. Geometry 121

Unit 8 Worksheet 4 1. Look at the coordinate grid below. Two points are to be added to the grid to form a square. a. Place two points in Quadrant 2 b. Place two points in Quadrant 3 that would form a square with that would form a square with the existing points. the existing points. Give the coordinates of the two points Give the coordinates of the two points. Complete the sketch of the square. Complete the sketch of the square. 2. Look at the coordinate grid below. Two points are to be added to the grid to form a rectangle with an area of 20 square units. a. Place two points in Quadrant 4 so b. Place two points in Quadrant 1 so that a rectangle with an area of that a rectangle with an area of 20 sq. units is formed. 20 sq. units is formed. Give the coordinates of the two points. Complete the sketch of the rectangle. Give the coordinates of the two points. Complete the sketch of the rectangle. 3. Look at the coordinate grid below. Point D is to be added in Quadrant 1to form a square. The slope of is and the slope of is. a. Use the slope information to help you plot point D to form a square. Complete the sketch of the square. Give the coordinates of point D. b. Use slopes to show that ACD has 4 right angles A c. Use the distance formula to show that all 4 sides in ACD are congruent. C (cont) Geometry 122

Unit 8 Worksheet 4 (cont) 4. The points (2, 1), (5, 1) and (2, 1) are plotted below. a. Plot a fourth point in quadrant 4 that will make a parallelogram. Give the coordinate of that point. Complete the sketch of the parallelogram. b. Plot a fourth point in quadrant 1 that will make a parallelogram. Give the coordinate of that point. Complete the sketch of the parallelogram. c. Plot a fourth point in quadrant 3 that will make a parallelogram. Give the coordinate of that point. Complete the sketch of the parallelogram. 5. The points ( 1, 1), ( 1, 3) and (1, 1) are plotted below. a. Plot a fourth point in quadrant 1 that will make a parallelogram. Give the coordinate of that point. Complete the sketch of the parallelogram. b. Plot a fourth point in quadrant 2 that will make a parallelogram. Give the coordinate of that point. Complete the sketch of the parallelogram. c. Plot a fourth point in quadrant 4 that will make a parallelogram. Give the coordinate of that point. Complete the sketch of the parallelogram. Geometry 123

Unit 8 Review In problems 1 22, choose the correct multiple choice response. NOTE: Diagrams are not drawn to scale. 1. What is the value of x? A. 540 120. 390 C. 150 D. 120 x 2. Determine the value of x. A. 15. 15.4 C. 9 D. 19.8 (8x + 1) 95 (5x 4) 73 3. Determine the sum of the exterior angles of an octagon. A. 1440. 1080 C. 360 D. 135 4. Determine the measure of each interior angle of a regular sided polygon with 9 sides. A. 1620. 180 C. 1260 D. 140 5. Determine the measure of each exterior angle of a regular polygon with 12 sides. A. 30. 150 C. 216 D. 36 6. The measure of an interior angle of a regular polygon is 162. How many sides does the polygon have? A. 18 sides. 20 sides C. 16 sides D. 10 sides 7. Determine the value of x? A. 80. 40 C. 60 D. 20 100 100 140 140 (cont) Geometry 124 100 x

Unit 8 Review (cont) 8. If the sum of the interior angles of a polygon equals 3780, how many sides does the polygon have? A. 23. 21 C. 20 D. 19 9. What is the measure of each interior angle of a regular hexagon? A. 60. 135 C. 45 D. 120 10. What are the values of the variables in the given parallelogram? A. x = 7, y = 9. x = 7, y = 65 C. x = 5, y = 71 D. x = 3, y = 77 11. If FH = 30, find FK. A. 12. 18 C. 15 D. 30 12. PQVT is a rhombus. Determine the value of x. A. 110. 55 C. 70 D. 35 T 13. If PQRS is a rhombus, which statement must be true? A. is a right angle. C. P D. F J 12 K (6x 8) G H P (2y + 16) 110 x V S (4x + 6) Q Q R 14. Which statement is true? A. All quadrilaterals are rectangles. All rectangles are quadrilaterals C. All rectangles are squares D. All quadrilaterals are squares (cont) Geometry 125

Unit 8 Review (cont) 15. Determine the value of x. A. 4. 6 C. 12 D. 8 2y 10 x + 6 3x 6 y + 2 16. In the diagram 1 = 9x, 2 = x + y Determine the values of x and y A. x = 20, y = 165. x = 10, y = 80 C. x = 20, y = 160 D. x = 10, y = 85 1 2 17. and are congruent base angles of isosceles trapezoid JKLM. If = (18x + 5), = (14x + 15) and = (17x + 10), determine the value of x. A.. 2 C. 15 D. 5 18. The perimeter of square MNOP is 72 inches, and NO = 2x + 6. What is the value of x? A. 15. 12 C. 6 D. 9 19. Determine the length of in the trapezoid shown. A. 26. 4 C. 13 D. 17 20. Which quadrilateral has two pairs of consecutive congruent sides, but opposite sides are not congruent? A. Kite. Rhombus C. Trapezoid D. Parallelogram (cont) N H K 4x + 1 5x + 2 27 L I M Geometry 126

Unit 8 Review (cont) 21. What is the most specific name for the figure shown? A. Quadrilateral. Parallelogram C. Trapezoid D. Rectangle 22. Determine the value of x in the given parallelogram. A. 2. 3 C. 4 D. 6 91 2x + y 4x + 3y 24 10 Work the following, showing all work. 2x + 7y 23. Determine the values of x and y. 7 11 3x + y 24. ACD is an isosceles trapezoid with midsegment. Determine the following: 10 (3x + 2) C n = EF = E 3n 4 F x = = A (2x 17) 36 D 25. ACD is a parallelogram. Determine the following. x = 3x + 2y C y = 7x y z 2 E z + 6 21 z = AC = A 26 D (cont) Geometry 127

Unit 8 Review (cont) 26. ACD is a rectangle with perimeter 96 meters. Determine the following. A = 28 E 2n + 8 = n = D 6n 12 27. Given a 25-gon: a. What is the sum of the measures of the interior angles? b. What is the sum of the measures of the exterior angles? C C = 28. Sketch LMNP if L (2, 1), M (1, 4), N (7, 6), and P (8, 3). Prove that LMNP is a rectangle. 29. The points (1, 1), (4, 1) and (1, 4) are plotted below. A. Plot a fourth point in. Plot a fourth point in C. Plot a fourth point in quadrant 4 that will make quadrant 3 that will make quadrant 1 that will a parallelogram. a parallelogram. make a parallelogram. Geometry 128

Unit 8 Systems Practice What values must x and y have to make each quadrilateral a parallelogram? 1. 2. (8x 6) o 3y o y 2 x 42 o (3x 40) ( y + 30) 3. 4. 42 (7y 2) o 26 3x 2y (4x + 1) o 4x + y 9y o 3x o Geometry 129

(5n + 1) Unit 8 Quadrilateral & Polygon Worksheet [1-12]: Solve for the variables. Give the best name for each of the following based upon given information and calculations. Show logical & appropriate work! {Names of Quadrilaterals are: Quadrilateral, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, & Isosceles Trapezoid} 1. 2. 1. x = 5x o 4y o y = 70 o 3. 4. Name: 2. x = y = Name 3. x = y = Name: 4. x = y = 5. 6. Name: 52 o x o x o y o y o 37 o (8n 11) 5. x = y = n = Name: 6. x = 8x 7 7. 8. Name Perimeter of quadrilateral is 90 cm 24 o 5n + 7 (7x 19) o y o (5x + 3) o 3n + 2 3x +1 A 115 o (12x 19) o (cont) y o 4n + 12 x o 2n + 40 Perimeter of quadrilateral is 274 meters. (x 4) feet C 15 feet 24 feet D 7. x = y = n = Name 8. x = C = Name Geometry 130

Unit 8 Quadrilateral & Polygon Worksheet (cont) 9. 10. 9. x = 3x + 5 C 2x 2 C AD = A 46 5x 1 D 2x + 1 (4a + 1) o 3x 3 (6a 13) o A Name = 10. x = z = Name 11. 12. 11. x = Quadrilateral ACD is a parallelogram. Quadrilateral ACD is a rhombus. 24 C 63 o 17 x o (2y + 1) o 7n + 3 y = m = n = 12. x = y = A D A D z = 5m 6 = 13. Find the sum of the measures of the interior angles for the following convex polygons. a. 17-gon b. 34-gon c. 51-gon 13.a. 14. Find the measure of each exterior angle for the following regular polygons. b. c. a. Pentagon b. Heptagon c. 45-gon 14.a. b. c. 15. Find the measure of each interior angle for the following regular polygons. 15.a. a. Decagon b. Octagon c. 21-gon b. c. 16. Find the number of sides for a convex polygon whose interior angle sum is: 16a. a. 3060 o b. 5400 o c. 4500 o b. 17. Find the number of sides for the following regular polygons, given: y o 25 o c. a. The measure of each exterior angle is 7.5 o. 17.a. b. The measure of each interior angles is 157.5 b. x o z o D C Geometry 131

Quadrilateral Tree Quadrilateral Kite 1. 1. Trapezoid 1. 2. 3. Parallelogram Isosceles Trapezoid 1. 2. 1. 2. 3. Rectangle 3. 4. 5. Rhombus Please note: Midsegment = average of the bases ase # 2 1. 2. Square 1. 2. 3. midsegment 1. ase # 1 or Geometry 132

Unit 8 Extra Practice In problems 1-3 find the requested parts. (Remember names of Quadrilaterals are: Quadrilateral, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, Isosceles Trapezoid, & Kite} 1. Perimeter = 144 meters x = x o y o 52 o 3n +13 y = n = est Name: 2. 8n 7. 2n 4 C x = = 2n + 5 n = = A (5x + 3) o 4n + 5 (3x + 35) o D est Name: 3. y o (10n 13) x = = x o n = (7n + 5) 65 o = est Name: 4. Find the measure of each interior angle for a regular 15-gon. 5. Find the measure of each exterior angle for a regular heptagon. 6. Find the number of sides for a regular polygon with each exterior angle measuring 15 o. 7. Find the number of sides for a regular polygon with each interior angle measuring 172 o. Geometry 133

Properties of Quadrilaterals Property Parallelogram Rectangle Rhombus Square Trapezoid Kite Two pairs of opposite sides are parallel. Has exactly one pair of parallel sides. Two pairs of opposite sides are congruent. Has two pairs of consecutive congruent sides, but opposite sides are not congruent. All sides are congruent. Diagonals are congruent. Diagonals are perpendicular. A diagonal bisects two angles. A diagonal forms two congruent triangles. Diagonals bisect each other. Opposite angles are congruent. All angles are right angles. Consecutive interior angles are supplementary. Geometry 134

Unit 10 Objective 0 1. A rhombus has a diagonal that lies on the line y = x + 1. What is the slope of the other diagonal of the rhombus? 2. Find the measure of in parallelogram ACD. Z A 40 3. LMNP is a rectangle. 2 and 3 are congruent. What is the measure of 1? L 1 D M C 50 P 2 3 4. In the diagram shown the measure of 1 is 4 times as large as the measure of 2. What is the measure of 2? N 2 1 5. Which multiple choice serves as a counterexample to the statement: All quadrilaterals have four right angles. A. A square. A rhombus C. A rectangle 6. In rectangle ACD, A =, CD =. determine the value of x. 7. PQRS is a rhombus. What is the value of y? x + 5 P (cont) S y 6 2x Q Geometry 135 S

Unit 10 Objective 0 (cont) 8. Currently the only way to travel from the city of Rio to the city of Phillie is by traveling through the city of Duke as shown in the drawing at the right. Engineers are thinking about building a road directly from the city of Rio to the city of Phillie as shown below. a. Determine the number of miles that the direct route frm Rio to Phillie would be. b. Determine how many miles would be saved by a person driving the direct route as compared to a person driving the long way through Duke. New Proposed Route Phillie 12 miles Phillie 12 miles Duke 5 miles Original Route Rio Duke 5 miles Rio 9. Given the parallelogram below, find the coordinates for P, without using any new variables. y (a, b) P (c, 0) x 10. For the quadrilateral shown, find W V 112 Y Z 47 Geometry 136

Geometry Unit 10 1. Parts of a circle, including tangent lines. (Section 10.1) 2. Central angles and finding arc measures. (Section 10.2) 3. Apply properties of chords. (Section 10.3) 4. Use inscribed angles and inscribed polygons. (Section 10.4) 5. Interior and exterior angle relationships with circles. (Section 10.5) 6. Find segment lengths in circles. (Section 10.6) 7. Write and graph equations of circles. (Section 10.7) Review Unit 10 Worksheet 4 asic Terms & Tangents In problems 1 6 refer to O. Name each of the following: 1. Two radii and 2. A diameter 3. A secant 4. A tangent 5. Two chords and 6. A point of tangency In problems 7 11 refer to with radius P. Find the following: 7. If P = 4, then SP = 8. If SP = 16n, then P = 9. If is tangent to, then =. 10. If and are tangent to, then. 11. If is tangent to, then would be to. (cont) Geometry 137

Unit 10 Worksheet 4 asic Terms & Tangents (cont) In problems 12 14 refer to O. 12. If = 60, then A = 13. If = 90, then C = 14. Name an inscribed polygon in the figure In problems 15 17, O and P are the centers of the circles. In problem 16, and are tangent to both circles and divides into segments whose lengths are shown. 15. 16. 17. OP = RS = HI = In the diagram for Problems 18 20, is tangent to O. 18. If DE = 12 and DO = 9, then OE = 19. If = 60 and OD = 9, then OE = 20. If DO = 5 and CE = 8, then DE = Geometry 138

Unit 10 Worksheet 5A In problems 1 4, and are chords. 1. If = 85 and = 73, then 1 =. 2. If = 136 and = 96, then 1 =. 3. If 1 = 54 and = 78, then =. 4. If 1 = 48 and = 42, then =. In problems 5 7, and are tangents. 5. If = 280, then =. 6. If = 96, then =. 7. If = 90, then =. In problems 8 10, is a tangent. 8. If = 120 and = 40, then =. 9. If = 45 and = 55, then =. 10. If = 50 and = 110, then =. In problems 11 15, and are secants. 11. If = 100 and = 20, then =. 12. If = 130 and = 40, then =. 13. If = 25 and = 25, then =. 14. If = 40 and = 130, then =. 15. If = 90, = 60, and = 80, then =. In problems 16 19, is tangent to the circle at point E. 16. If = 100 and = 20, then =. 17. If = 25 and = 25, then =. 18. If = 95 and = 25, then =. 19. If = 40 and = 138, then =. Geometry 139

In problems 1 6, find the values of a, b, and c. Unit 10 Worksheet 5 1. 2. a = b = c = a = b = c = 3. 4. a = b = c = a = b = c = 5. 6. a = b = c = a = b = c = is a diameter of O. is tangent to O at A. = 80, = 20, and = 50. 1 = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = 10 = Geometry 140

Unit 10 Worksheet 6A Solve for x. Show your work. 1. 2. 3. 4. 5. 6. x 7. 8. 9. 10. 11. 12. 13. (cont) Geometry 141

Unit 10 Worksheet 6A (cont) 14. 15. 16. 17. 18. 19. 20. 21. A = 48 Find the radius. Given: Circle O = 124, = 140, = 62. Find the measure of 1 through 10 Geometry 142

Unit 10 Worksheet 6 In O, = 50 and = 70. Find each of the following: 1. 2. 3. 4. 5. 6. In O, = 60, = 80 = 110, is tangent to O at C, and = 130. Find each of the following: 7. 1 8. 2 9. 10. 11. 3 12. O has arc measures as shown. and are tangent at J and M, respectively. Find the following: 13. 14. 1 15. 2 16. 3 (cont) Geometry 143

Unit 10 Worksheet 6 17. is tangent to O at Q. If = 15 and PO = 17, find the radius of the circle. 18. Find the total number of common tangents that can be drawn to two coplanar circles that are externally tangent. 19. Complete: If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is. In O, 1 2 and 3 = 100. Find the following: 20. 21. 22. 23. In P, = 300. Find the following: 24. 25. 26. AD = 27. D = 28. 29. X PQ = 16; OX = 6 ; OX = 5 OY = 7; RS = GH = 24; OG = 30. Find the length of a chord that is 3 cm from the center of a circle with radius 9 cm. Geometry 144

Unit 10 Worksheet 7 Solve for x. Show all work. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. (cont) Geometry 145

Unit 10 Worksheet 7 12. 13. 14. 15. 16. 17. 18. 19. 20. x 2 + (y 4) 2 = 72 21. Center = (5, 1), radius = Center = Write the equation for the circle Radius = given the above information. Geometry 146

Unit 10 Review In problems 1 17 select the correct multiple choice response 1. EF and EG are tangents to the circle shown. Find the measure of. A. 30. 15 C. 60 D. 12 6. Find the value of x. A. 8. 15 C. 22 D. 17 2. HJ is tangent to P. Find the value of x. A. 18. 24 C. 25 D. 32 3. Find in P A. 290. 178 C. 288 D. 292 4. Find. A. 166. 173 C. 152 D. 208 7. In P the 42. Find A. 42. 48 C. 96 D. 84 8. WY is tangent to the circle shown. = 74. Find. A. 286. 148 C. 212 D. 323 9. Find the value of x in the circle shown. A. 54. 27 C. 49 D. 76 5. Find the value of x. A. 40. 20 C. 60 D. 30 10. Find the value of x in the circle shown. A. 36. 52 C. 88 D. 46 Geometry 147

11. What is the length of? 266 A. 5. 7 C. 12 D. 14 17. Find A. 52. 78 C. 64 D. 32 12. Find CD. A. 2. 24 C. 9 D. 18 Solve the following problems. Show all work. 18. C and AC are tangents to the given circle. Find the value of x. 13. In the circle shown, UP = 2, NP = 4, and UW = 18. Find LP. A. 9. 8 C. 12 D. 13 19. If = 110, find 14. Find in O. A. 78. 90 C. 156 D. 34 20. In Q = 220 a. Find b. Find 15. State the radius of the circle whose equation is (x 1) 2 + (y 3) 2 = 4 A. 4. 16 C. 8 D. 2 21. Find the value of x. 16. State the center of the radius of the circle whose equation is (x + 6) 2 + (y 7) 2 = 1 A. (6, 7). (6, 7) C. ( 6, 7) D. ( 6, 7) 22. Circle O is inscribed in quadrilateral ACD. A = 12 and CD = 13. 4 Find the perimeter of quadrilateral ACD. 6 C D Geometry 148 A

Problems 1-4 refer to Unit 10 Worksheet Arc, Central Angles, and Chords O. Find the measure of each arc. 1. = 2. = 3. = 4. = Find the value of x. Each angle shown is a central angle. 5. 6. 7. x = x = x = 8. At 10 o clock the hands of a clock form an angle of. 9. At seven o clock the hands of a clock form an angle of. 10. If the hands of a clock form an angle of 30, the time is o clock. In problems 11 16, is a diameter of O. 11. E = 12. O = 13. = 14. = 15. = 16. DE = Complete the following: 17. A = 8, CD = 9 ED = 18. HI = 19. WY = 20. CD = Geometry 149

Unit 10 Circles & Special Right Triangles 1. Find A 2. Find A 3. is tangent to A C C Find A 5 A A 4. Find the value of x, y, and 5. OT = 9, RS = 18, 6. = 90 and XZ = 13 Find OR Find XY 7. Find JK 8. Find LM and LO 9. Find 10. Find 11. Find K and 12. Find SU and Geometry 150