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MP4056 Grades 6-8 E F D Q J N H G P 6.5cm 6 8 10 12 5cm 6cm 5cm Milliken Publishing Co. • St. Louis, Missouri Geometry Grades 6–8 Pythagoras would be proud! The theorems and principles of basic geometry are clearly presented in this workbook, along with examples and exercises for practice. Author Janice Wendling Artist All concepts are explained in an easy–to– understand fashion to help students grasp geometry and form a solid foundation for advanced learning in mathematics. Elizabeth Adams Editing and Page Design Martha Kranes Cover Design Gray Communications & Marketing Each page introduces a new concept, along with a puzzle or riddle which reveals a fun fact. Thought–provoking exercises encourage students to enjoy working the pages while gaining valuable practice in geometry. Project Director Kathleen Hilmes Copyright © 1995 Milliken Publishing Company 11643 Lilburn Park Dr. St. Louis, MO 63146 All rights reserved. www.millikenpub.com The purchase of this book entitles the individual purchaser to reproduce copies by duplicating master or by any photocopy process for single classroom use.The reproduction of any part of this book for commercial resale or for use by an entire school or school system is strictly prohibited. Storage of any part of this book in any type of electronic retrieval system is prohibited unless purchaser receives written authorization from the publisher. Geometry Workbook Table of Contents Reading Mathematics . . . . . . . . . . . . . . . . . . .1 Undefined Terms and Basic Definitions . . . . .2 Types of Angles . . . . . . . . . . . . . . . . . . . . . . .3 Complementary and Supplementary Angles . . .4 Parallel, Perpendicular, and Skew Lines . . . .5 Angles Formed by Parallel Lines . . . . . . . . . .6 Angle Sum Theorem . . . . . . . . . . . . . . . . . . . .7 Exterior Angle Theorem . . . . . . . . . . . . . . . . .8 Classifying Triangles . . . . . . . . . . . . . . . . . . . .9 The Pythagorean Theorem . . . . . . . . . . .10-11 The Converse of the Pythagorean Theorem . . .12 Congruent Figures . . . . . . . . . . . . . . . . . . . .13 Congruent Triangles . . . . . . . . . . . . . . . . . . .14 Congruent Triangles . . . . . . . . . . . . . . . . . . .15 Angle and Triangle Word Search . . . . . . . . .16 Types of Quadrilaterals . . . . . . . . . . . . . . . . .17 Properties of Parallelograms . . . . . . . . . . . . .18 Properties of Rectangles, Rhombuses, and Squares . . . . . . . . . . . . . . . . . . . . . . . .19 Trapezoids . . . . . . . . . . . . . . . . . . . . . . . . . .20 Ratio and Proportion . . . . . . . . . . . . . . . . . . .21 Similar Figures . . . . . . . . . . . . . . . . . . . . . . .22 © Milliken Publishing Company Trigonometric Ratios . . . . . . . . . . . . . . . . . . . . . .23 Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . .24 Area of Rectangles and Triangles . . . . . . . . .25 Area of Trapezoids . . . . . . . . . . . . . . . . .26-27 Area of Parallelograms . . . . . . . . . . . . . .28-29 Word Search . . . . . . . . . . . . . . . . . . . . . . . . .30 Circumference . . . . . . . . . . . . . . . . . . . . . . . .31 Area of a Circle . . . . . . . . . . . . . . . . . . . . . . .32 Area of Irregular Shapes . . . . . . . . . . . . . . . .33 Area of a Shaded Region . . . . . . . . . . . . . . .34 Three–Dimensional Figures . . . . . . . . . . . . .35 Surface Area of Right Prisms . . . . . . . . . . . .36 Volume of Rectangular Prisms . . . . . . . . . . .37 Volume of Right Prisms . . . . . . . . . . . . . . . .38 Surface Area of Regular Pyramids . . . . . . . .39 Volume of Regular Pyramids . . . . . . . . . . . .40 Surface Area of Cylinders . . . . . . . . . . . . . . .41 Surface Area of Cones . . . . . . . . . . . . . . . . .42 Volume of Cylinders and Cones . . . . . . . . . .43 Crossword Puzzle . . . . . . . . . . . . . . . . . . . . .44 Answer Key . . . . . . . . . . . . . . . . . . . . . . .45-48 MP4056 Reading Mathematics Remember: Learning the correct meaning and use of mathematical symbols is necessary for reading and understanding mathematics. In geometry, the order of the letters is important in some cases, like when naming rays and angles. Find the corresponding symbols and shade their areas to reveal a number that is neither prime nor composite. 1. line AB 14. angle with vertex at C 2. segment AB 15. cosine of angle X 3. angle with vertex at B 16. is approximately equal to 4. triangle ABC 17. is similar to 5. cube root 18. greater than 6. arc AB 19. square root 7. ray AB 20. is congruent to 8. circle with center A 21. pi 9. tangent of angle X 22. is perpendicular to 10. not equal to 23. parallelogram ABCD 11. is parallel to 24. sine of angle X 12. right angle 25. less than 13. measure of angle A 26. ordered pair © Milliken Publishing Company 1 MP4056 Undefined Terms and Basic Definitions Geometry is based on the undefined terms point, line, and plane. Points can be collinear (lie on the same line). Points and lines can be coplanar (lie in the same plane). A ray is part of a line with one endpoint. An angle is formed by two rays that have the same endpoint. Refer to the diagrams and determine whether the statement is true or false. If it is true, place its corresponding letter in the puzzle to reveal the name of a famous mathematician and his collection of books about geometry, number theory, and geometric algebra. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. CK intersects RL at O. E M lies in plane X. A R, O, K, and A are coplanar. R OR and OK are sides of ∠ROK. U A, O, and B are collinear. C O, L, K, and M are coplanar. H C, O, A, and B are coplanar. L Plane D contains P. P Plane X intersects AB at O. I A, B, and M are coplanar. D Plane X contains Z. S J, N, F, and P are coplanar. A ∠CBO lies in plane X. K 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. NJ and NH are opposite rays. ZL intersects RO at O. GF intersects JH at N. E F L Plane E contains ∠JNF. I Plane D and Plane E intersect in GF. E G, N, F, and P are coplanar. M Q, H, N, and F are coplanar. E N and P are in plane D. I ∠GNH lies in plane D. N G, N, and F are collinear. T Plane D contains PQ. B NF and NG are opposite rays. S E A X F K R D O Q L C N J H Z G M B P © Milliken Publishing Company 2 MP4056 Types of Angles An acute angle measures between 0° and 90°. A right angle measures exactly 90°. An obtuse angle measures between 90° and 180°. A straight angle measures exactly 180°. Refer to the diagram and classify each expression as acute, right, obtuse, or straight. 1. ∠CDL __________________________ 11. ∠EDA + ∠ADL _____________________ 2. ∠EAD __________________________ 12. ∠ECD + ∠DCL _____________________ 3. ∠DAB __________________________ 13. ∠ADE + ∠EDC _____________________ 4. ∠CLD __________________________ 14. ∠UEC + ∠CEL _____________________ 5. ∠ABI ___________________________ 15. ∠EDC + ∠CDL _____________________ 6. ∠CLI ___________________________ 16. ∠ABD + ∠DBL _____________________ 7. ∠UCL __________________________ 17. ∠CDE + ∠EDB _____________________ 8. ∠ABL __________________________ 18. ∠EDA + ∠ADB _____________________ 9. ∠ECL __________________________ 19. ∠EDA + ∠ADB + ∠EDC ______________ 10. ∠AED __________________________ 20. ∠LDB + ∠BDA + ∠ADE ______________ E A B I D U © Milliken Publishing Company C L 3 MP4056 Complementary and Supplementary Angles Remember: alphabetic order complementary supplementary numeric order 90° 180° ADD TO Use a ruler to match each angle to its complement and its supplement. Each line you draw will cross a letter. The letters without lines through them spell the answer. Michael Jordan scored his career 20,000th point in Chicago. In what cities did he score his 5,000th, 10,000th, and 15,000th points? Complement Angle A 47° 63° Supplement 52° H C L C 78° 38° C G 13° L M 71° C H 117° K 43° 27° A D L 83° 79° 19° 27° O H 146° 153° T 34° 103° H 11° C R W L P P 22° 13° C 68° L 9° 158° N T 68° D 7° I 56° E 161° G G 77° 102° 128° 63° H 167° 137° D 12° U 81° I O 77° I 22° ____ ____ ____ O R 173° 169° T A 112° O 99° ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ © Milliken Publishing Company 4 MP4056 Parallel, Perpendicular, and Skew Lines Parallel lines are coplanar lines that never intersect. Skew lines are noncoplanar lines (neither parallel nor intersecting). Perpendicular lines intersect to form right angles. Planes can also be parallel, perpendicular, or intersecting. pa ral lel pe rpe nd icu sk lar ew int ers ec tin g pa ral lel pe rpe nd icu sk lar ew int ers ec tin g Determine whether the following pairs of lines are parallel, perpendicular, skew, or intersecting (not perpencicular). The figure is a cube—all faces are squares. Place the correct corresponding letter in the blanks below to reveal the mathematician who developed hyperbolic geometry. 1. AB and FG N P A D 10. HC and AB B F O M 2. AG and PQ E B J I 11. GD and AB I T A X 3. FE and FG Z K Q A 12. FG and GD L C F D 4. CD and HE O C V B 13. FE and BC H A E D 5. AG and HD L E M K 14. AB and AG R D L E 6. PQ and HD O Y A R 15. GB and DE N Q V P 7. AH and HC P I T W 16. EH and DH U W G S 8. HD and JK T U Y L 17. HC and BC K K I L 9. FG and BC H E O B 18. AG and JK D C Y H A B P X ____ ____ ____ ____ ____ ____ ____ F 1 G 2 3 4 5 6 7 Q J H C ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 8 9 10 11 12 13 14 15 16 17 18 Y E K © Milliken Publishing Company D 5 MP4056 Angles Formed by Parallel Lines If two parallel lines are cut by a transversal, the resulting angles will either be congruent or supplementary. congruent angles vertical angles corresponding angles alternate interior angles alternate exterior angles supplementary angles adjacent angles same–side interior angles same–side exterior angles Determine whether the angles listed are conguent, supplementary, or neither. If the angles are congruent, write the corresponding letters in the top box below. If the angles are supplements, write the corresponding letters in the lower box below. 1. ∠1 and ∠5 I 11. ∠4 and ∠7 F 21. ∠2 and ∠13 I 2. ∠1 and ∠6 G 12. ∠11 R 22. ∠4 and ∠8 W 3. ∠1 and ∠3 E 13. ∠5 and ∠6 C 23. ∠9 and ∠14 B 4. ∠5 and ∠10 O 14. ∠7 and ∠12 I 24. ∠1 and ∠13 T 5. ∠3 and ∠16 T 15. ∠10 and ∠14 N 25. ∠3 and ∠8 N 6. ∠7 and ∠11 S 16. ∠10 and ∠13 E 26. ∠3 and ∠15 O 7. ∠4 and ∠16 A 17. ∠1 and ∠14 D 27. ∠12 8. ∠6 and ∠9 T 18. ∠8 and ∠12 E 28. ∠5 and ∠9 N 9. ∠9 and ∠12 U 19. ∠2 and ∠5 L 29. ∠6 and ∠7 D 10. ∠6 and ∠10 A 20. ∠8 and ∠11 E 30. ∠4 and ∠15 Z and ∠16 The creators of calculus were: (congruent) and ∠15 l k __ __ __ __ __ __ __ __ __ 1 __ __ __ __ __ __ __ I 2 5 6 m 3 4 7 8 m || n AND (supplementary) 9 10 13 14 __ __ __ __ __ 11 12 15 16 n __ __ __ __ __ __ © Milliken Publishing Company 6 MP4056 Angle Sum Theorem The three angles of a triangle add to 180°. Find the missing angle measure for each triangle. Use the decoder to read the message below. DECODER 1. 25°, 65°, _______ A = 90° 2. 42°, 120°, _______ C = 111° 3. 37°, 90°, _______ E = 135° 4. 23°, 46°, _______ F = 95° 5. 50°, 105°, _______ G = 24° 6. 136°, 20°, _______ H = 100° 7. 15°, 30°, _______ I 8. 30°, 45°, _______ L = 93° 9. 52°, 35°, _______ M = 86° 10. 67°, 13°, _______ N = 18° 11. 74°, 100°, _______ O = 6° 12. 9°, 110°, _______ P = 130° 13. 24°, 26°, _______ R = 25° 14. 10°, 84°, _______ S = 61° 15. 17°, 25°, _______ T = 105° 16. 40°, 45°, _______ U = 138° 17. 53°, 35°, _______ Y = 92° ____ ____ ____ 8 10 ____ ____ ____ ____ ____ 7 1 4 15 ____ ____ 11 8 ____ 16 ____ ____ ____ ____ ____ ____ 7 1 2 5 3 1 6 9 7 12 ____ ____ ____ ____ ____ 1 5 3 6 ____ ____ ____ ____ ____ ____ ____ ____ 8 = 53° 2 6 9 10 8 ____ ____ ____ 7 1 5 7 ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 4 11 © Milliken Publishing Company 14 13 9 7 14 7 7 2 8 1 5 17 MP4056 Exterior Angle Theorem In any triangle, the measure of one exterior angle is equal to the sum of its remote interior angles. An exterior angle and its adjacent interior angle are supplementary. 2 ∠1 + ∠2 = ∠4 ∠3 + ∠4 = 180° ∠1 + ∠2 + ∠3 = 180° 3 4 1 1. 2. y 3. x y y x = _____ y = _____ 4. x z x = _____ z = _____ y = _____ 5. x x = _____ 6. x y = _____ x z w z x y y x = _____ 7. x y y = _____ w = _____ y = _____ x = _____ z = _____ y w = _____ y = _____ w x = _____ z = _____ z x = _____ z = _____ © Milliken Publishing Company y = _____ 8 MP4053 Classifying Triangles by sides by angles equilateral — all three sides equal isosceles — at least two sides equal scalene — no sides equal equiangular — all three angles equal acute — three acute angles right — one right angle obtuse — one obtuse angle Classify the triangles below. There will be at least 2 answers per triangle, maybe more. Check all columns that apply in the chart. Write the letters corresponding with those columns checked in the blanks below to reveal the “Prince of Mathematicians”. 1. 2. 3. 15 8 6 5 4 10 10 10 3 4. 6. 5. 7 7 eq ui la te ra iso l sc el es sc al en e eq ui an gu la ac r ut e 7 rig ht ob tu se 7. 1. E L C U D A F 2. B R N O C I L 3. N W F T S R P 4. A C I G E Y H 5. D R P I C S W 6. V M H Q E D G 7. R A E D H U I 8. P S M L S N C 4 4 8. 5 5 ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ © Milliken Publishing Company 9 ____ ____ ____ ____ ____ MP4056 The Pythagorean Theorem In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs. c a The hypotenuse is opposite the right angle. The legs form the right angle. b 2 2 2 Pythagorean Theorem: a + b = c Solve for the missing side. Match your answer in the decoder to find the special name for three integers whose lengths form a right triangle. 1. 2. 6 3. 5 8 15 8 12 6. 5. 4. 8 4 10 12 3 9 7. 7 © Milliken Publishing Company 24 10 MP4056 8. 9. 10. 15 10 4 17 26 5 11. 12. 13. 16 15 15 20 25 12 A E G H 17 5 15 3 I L 10 24 N 9 O 20 P 8 R S T Y 25 12 13 6 ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 9 6 2 8 3 5 13 7 4 3 12 ____ ____ ____ ____ ____ ____ ____ 2 © Milliken Publishing Company 7 1 9 11 10 4 11 MP4056 The Converse of the Pythagorean Theorem The Pythagorean Theorem can also be used to determine whether any triangle is acute, right, or obtuse. I. Find two shorter sides, square each, add the squares. II. Square the longest side. III. Compare the results: longest side2 < short side2 + short side2 ⇒ ACUTE longest side2 = short side2 + short side2 ⇒ RIGHT longest side2 > short side2 + short side2 ⇒ OBTUSE OB TU SE LENGTHS RI GH T AC UT E Determine whether the following lengths create an acute, right, or obtuse triangle. Check the corresponding column in the chart and write its letter in the blanks below. 1. 3, 4, 5 A T D U.S. Olympian Mike Conley won 2. 9, 9, 13 I P H a silver medal in the 1984 3. 11, 11, 15 E S M Summer Olympics and a gold medal 4. 7, 7, 7 T A C for the same event in the 1992 5. 6, 8, 10 U R H Summer Olympics. In which 6. 8, 10, 12 I L S event did he compete? 7. 4, 6, 8 A Z P 8. 5, 12, 13 M L N 9. 8, 14, 17 T I E 10. 6, 7, 8 J H O 11. 9, 12, 15 E U L 12. 13, 14, 15 M O N 13. 7, 8, 11 S U P © Milliken Publishing Company ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 12 MP4056 Congruent Figures Two figures are congruent if they are the same size and shape. Congruent parts of the figures (segments and angles) are called corresponding parts. Name the corresponding parts for the congruent figures below. Remember that the order of the letters is important. 1. L 2. P T K M N R a. b. c. d. e. f. g. 3. ΔLMN ≅ Δ ______________________ ML ≅ __________________________ MN ≅ _________________________ ∠L ≅ ∠ ________________________ PS ≅ __________________________ ∠R ≅ ∠ ________________________ ∠N ≅ ∠ ________________________ D A a. b. c. d. e. f. g. L D I R S C B U N TRUCK ≅ ______________________ KC ≅ _________________________ TR ≅ _________________________ CU ≅ _________________________ UR ≅ _________________________ TK ≅ _________________________ ∠C ≅ ∠ ________________________ Q P T R a. b. c. d. DART ≅ ___________ TR ≅ ______________ QA ≅ _____________ ∠T ≅ ∠ _______________ © Milliken Publishing Company e. f. g. h. PQ ≅ ______________ RP ≅ ______________ ∠DAR ≅ ∠ ____________ ∠TRA ≅ ∠ ____________ 13 i. PRAQ ≅ ____________ j. TDAR ≅ _____________ k. RPQA ≅ ____________ MP4056 Congruent Triangles Three Methods of Proving Triangles Congruent 3 sides of one Δ ≅ to 3 sides of another Δ. 2 sides and the included angle of one Δ ≅ to 2 sides and the included angle of another Δ. 2 angles and the included side of one Δ ≅ to 2 angles and the included side of another Δ. Side—Side—Side (SSS) Side—Angle—Side (SAS) Angle—Side—Angle (ASA) Determine which of the above methods can prove the triangles congruent. If SSS, place an X in the left letter box; if SAS, place an X in the middle letter box; if ASA, place an X in the right letter box. The remaining letters will spell the name of a U.S. president who developed a proof of the Pythagorean Theorem. 3. 2. 1. RME J AB 4. S P A 5. 6. S A M B R E 7. G L A 8. R F L 9. I E A ____ ____ ____ ____ ____ L T D ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ © Milliken Publishing Company 14 MP4056 Congruent Triangles Two Additional Triangle Congruence Methods Angle—Angle—Side (AAS) Hypotenuse—Leg (HL) Use the following box as a guideline. Two angles and a nonincluded side of one Δ ≅ to two angles and a nonincluded side of another Δ. In a right Δ, the hypotenuse and one leg ≅ to the hypotenuse and leg of another right Δ. SSS SAS ASA AAS HL Place an X in all the boxes that can be applied to prove the triangles congruent.The remaining letters will reveal the name of a process used by Archimedes to determine the volume of a sphere using equal weights. 1. 3. 2. M P E T H T H A M E 4. 5. T H E Q U 7. O D O F R 6. S T I L I B R U S I 8. M T H O D U H E A P ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ © Milliken Publishing Company 15 MP4056 Angle and Triangle Word Search Find the words listed below in the puzzle. S H G D O P T K V H D J D D X S M A R R O T C E S I B K B R E U E O P P O S I T E G G E M L A N R U R A L U G N A I U Q E S B E S N J S T R A I G H T C A M I N E H M A T S K L A N S G A R O X O D R S T E U A S O S I E K T E O S T H H S B C S R H V D A L T I P X W G O L I O S B E Y E L R O E Q U I L A T E R A L V T S E E R D P R Z D R A F S J L E U V M P Y V E S J E S U E F S V P I U E E T U C A V D W R U B C P C C N J H W A C N M A N R K B L R U D E G G L E E M E A N D A E V E I H W A E N I T N V E U K M X F C E G K N T O B T U S E Y E A D U E K G E P N M K R A N E N S N L A T H Y A N S M A I L A T N D A E R H U S A R S H U L L A C Y R A T N E M E L P M O C H R D W I V K T S A I W A L K E R Y A obtuse acute scalene right equilateral equiangular isosceles bisector © Milliken Publishing Company straight sides complementary supplementary 16 vertical hypotenuse leg adjacent opposite perpendicular base vertex MP4056 Types of Quadrilaterals Quadrilateral Trapezoid (only one pair parallel sides) Parallelogram (two pairs of parallel sides) Rectangle (four right angles) Rhombus (all sides equal) Square Check all terms that apply for the shapes below. Remember, more than one column may be checked for a shape. Write the remaining letters in the blanks below to answer the question. 4 3 8 9 1. 2. 3. 4. 5 5 7 5 8 5. 3 6. 7. 6 2 9. 5 4 7 7 5 8 7 5 3 W I N T E N W O E 4 8. 5 5 3 quadrilateral 4 9 5 1. 2. 3. 4. 5. 6. 7. 8. 9. 6 6 10 10 4 6 12 10 trapezoid parallelogram A S L R A W A R G rectangle rhombus H A I N O I A H T L C F R H I I I O S K G O G A W S G square A A O I A W I N N Which four U.S. states have active volcanoes? ____ ____ ____ ____ ____ ____ , ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ , and ____ ____ ____ ____ ____ ____ ____ ____ ____ ___ © Milliken Publishing Company 17 MP4056 Properties of Parallelograms Parallelograms have all of the following properties: —both pairs of opposite sides parallel —both pairs of opposite sides congruent —both pairs of opposite angles congruent —diagonals bisect each other Refer to the diagram to determine the following measures. Place the answers in the cross–number puzzle. I K 2 3 4 1 S 78 E ACROSS 5 6 T 1. If KI = 12, then ET = ______________________. 2. If KE = 23, then IT = ______________________. 3. If ∠KET = 150° and ∠3 = 70°, then ∠4 = ______°. 4. If ES = 7, then EI = ______________________. DOWN 5. If ∠3 = 60° and ∠4 = 70°, then ∠ITE _________°. 1. If ∠IKE = 55°, then ∠KET = ________°. 6. If KT = 36, then ST = _____________________. 2. If KT = 48, then KS = ______________. 8. If ∠1 = 30° and ∠2 = 40°, then ∠KET = _______°. 3. If ∠EKI = 85 °, then ∠ETI = ________°. 9. If ∠EKI = 45°, then ∠KIT = ________________°. 6. If SI = 5, then EI = ________________. 10. If KS = 11.5, then KT = ___________________. .... .... .... .... 3 2 1 .... .... .... .... .... .... .... .... .... .... .... .... 4 .... .... .... ............................ .... .... .... .... 5 6 7 .... .... .... .... .... .... .... .... .... .... .... .... .... .... 8 .... .... .... .... ........ .... ........ .... .... .... .... .... 9 10 .... .... .... .... .... .... .... .... .... .... .... ........ ........ .... .... .... .... .... .... ........ ........ .... © Milliken Publishing Company 18 7. If ∠1 = 30° and ∠2 = 40°, then ∠5= __________°. 8. If ∠1 = 20° and ∠7 = 10°, then ∠KSE = __________°. 9. If KS = 5, then KT = _______________. 10. If ∠EKI = 25°, then ∠ETI = _________°. MP4056 Properties of Rectangles, Rhombuses, and Squares Rectangles all properties of parallelograms plus —diagonals are congruent —all angles measure 90° Rhombuses all properties of parallelograms plus —all sides are congruent —diagonals are perpendicular —diagonals bisect opposite angles Squares —all properties of parallelograms —all properties of rectangles —all properties of rhombuses Use the properties to find measures of segments and angles in the diagrams. 1. ABCD is a rectangle. If AB = 24, BC = 10, and ∠1 = 50°, find the following: A 1 2 3 B 4 d. BD = ______ g. ∠DAB = ______ e. AX = ______ h. ∠3 = ________ f. BX = ______ i. ∠AXB = ______ a. CD = ______ b. AD = ______ c. AC = ______ X C D 2. ABCD is a rhombus. If AB = 6, XC = 3, and ∠DAB = 120°, find the following: a. b. c. j. BC = _________ d. ∠AXB = _____ g. ∠3 = _____ = ______ e. ∠1 = _______ h. ∠4 = _____ = ______ f. ∠2 = _______ i. AX = _____ ΔABC is an _____________________________triangle. ∠ADC ∠DCB A 3 1 2 B 4 X D C 3. ABCD is a square. If AB = 16 and AC = 16 2 , find the following: A B 1 2 3 4 a. b. c. d. BC = ________ BD = ________ AD = ________ ∠1 = _________ e. f. g. h. ∠2 = ___________ ∠AXB = _________ ∠BXC = _________ ∠4 = ___________ X D © Milliken Publishing Company C 19 MP4056 Trapezoids All trapezoids have exactly one pair of parallel sides (called bases). An isosceles trapezoid has congruent legs, base angles, and diagonals. A right trapezoid has two right angles. Any trapezoid can be divided into a rectangle and triangle(s) to help find measurements of the sides and angles. To divide the trapezoid, draw altitudes between the bases. right trapezoid isosceles trapezoid general trapezoid Use the properties of trapezoids, rectangles, and right triangles to find the missing measures in the diagrams below. Remember to use the Pythagorean Theorem in right triangles. 6 1. 7 2. 5 10 6 4 4 z 10 5 6 6 y x 20 b 4. 4 g 5. f c 3 d a 3. 9 6 6. e 12 10 5 7. 25 8 5 j 4 13 12 12 13 k m © Milliken Publishing Company 20 MP4056 Ratio and Proportion An equation that sets two ratios equal to one another is a proportion. Use the means–extremes property to solve a proportion. The product of the means is equal to the product of the extremes. a:b=c:d a = c means OR b d extremes b and c are means a and d are extremes bc = ad bc = ad Solve the proportions below. Use the decoder to find the name of a special ratio and where it is found in ancient times. 1. 7 = 21 2. 3x = 12 3. x + 5 = 1 4 x 6 4 4 2 4. x + 2 = 4 x+3 5 5. 18 = 9 x 4 6. 3 = 18 4 x 7. 36 = 18 x 8 8. x = 84 9 108 9. x – 3 = 4 x 5 10. x = 6 6 4 11. 7x – 5 = x + 9 4 3 12. 13. x + 6 = x 16 10 14. x + 9 = 5x 2 1 15. 4x + 2 = 18 6x + 13 17 16. x = 9 12 4 17. 9 = 3 4x 1 18. 2x + 1 = 4x – 3 18 21 19. x – 1 = 6 x 8 20. 2x + 4 = 13 3x – 3 15 21. 5 = x 8 40 A B C D 15 24 16 12 E 2 ____ ____ ____ 11 13 4 ____ ____ ____ 3 9 8 ____ ____ ____ 11 13 4 ____ ____ ____ ____ 9 2 8 10 F G H I K 1 34 10 –5 4 M 5 2 N O P 11 9 27 ____ ____ ____ ____ ____ ____ 17 10 2 1 4 20 ____ ____ ____ ____ 21 8 4 1 ____ ____ 11 10 R 5 ____ ____ ____ ____ ____ 17 12 4 4 19 S 7 T U W 3 25 –3 Y 8 ____ ____ ____ ____ ____ 12 9 11 15 10 ____ ____ ____ ____ ____ ____ 1 4 8 15 17 20 ____ ____ ____ ____ ____ ____ ____ ____ 16 5 12 9 18 15 1 8 ____ ____ ____ 9 20 1 © Milliken Publishing Company L 6 5 = 15 x–2 x+4 ____ ____ ____ 9 20 1 ____ ____ ____ ____ ____ ____ ____ ____ ____ 6 21 15 2 1 15 20 17 8 ____ ____ ____ ____ ____ ____ ____ ____ ____. 9 12 11 15 14 9 7 11 8 21 MP4056 Similar Figures Figures are called similar when their corresponding sides are all in proportion and their corresponding angles are congruent. A D 7 7 14 B 7 14 C E ΔABC ~ ΔDEF F 14 ratio of sides is 7 : 14 or 1 : 2. Use proportions to solve for the missing sides in the similar figures below. 1. 2. 4 a y 8 8 4 8 x 3. 5 b 4. 8 6 12 3 6 6 c e 8 f 8 g 10 i 18 h d 5. 4 r 4 6 6. s 12 6 5 13 t v w u 7 39 14 © Milliken Publishing Company 22 MP4056 Trigonometric Ratios Chief SOHCAHTOA says, “Use my name to remember the three basic trigonometric ratios.” Opposite Sine = Hypotenuse Adjacent Cosine = Hypotenuse Opposite Tangent = Adjacent Use the triangles below to match the trigonometric ratios. Your answers will help you to find the name of the first woman in American history to receive a patent. A V 8 D E 12 S R X 25 4 5 B 17 C 3 5 15 1. sin ∠D 6. tan ∠C 11. sin ∠A 16. cos ∠W cos ∠Y tan ∠F tan ∠Z tan ∠D 2. 7. 12. 17. 13 9 W T F 24 tan ∠A sin ∠C tan ∠V sin ∠Y 3. 8. 13. 18. U 7 4. 9. 14. 19. Y sin ∠R cos ∠R cos ∠D tan ∠W 5. cos ∠V 10. tan ∠T 15. tan ∠R 20. tan ∠Y 9 41 7 25 12 13 9 40 24 25 3 4 4 3 5 12 15 17 12 5 7 24 5 13 15 8 40 9 4 5 A C D E F G H I K M N O P R S T 1 7 17 ____ ____ ____ ____ ____ 18 16 6 14 13 ____ ____ ____ ____ ____ ____ ____ ____ 17 9 2 9 6 ____ ____ ____ 12 13 19 9 ____ ____ ____ 17 3 9 © Milliken Publishing Company 7 2 3 5 V W 8 17 40 41 X Y 6 9 20 ____ ____ ____ ____ ____ ____ 4 7 8 9 10 8 ____ ____ ____ ____ ____ ____ ____ 17 3 15 7 11 9 7 ____ ____ ____ ____ ____ ____ ____ 1 24 7 ____ ____ ____ ____ 10 ____ 16 Z 41 8 15 ____ ____ ____ ____ 40 6 10 23 9 19 6 ____ ____ 6 10 5 1809. 10 MP4056 Perimeter Perimeter is the distance around a figure. Simply add the lengths of all sides that outline the figure to find its perimeter. Perimeter is measured in linear units, such as inches, centimeters, yards, meters, and so on. Find the perimeter of each figure below. Shade the answers below to reveal the number of bones in the human body. 2. 3. 1. 15 cm 8 in 6m 8 in 4 cm 8m 7 in 4. 5. 10 ft 6. 9m 15 cm 13 cm 9m 5 ft 5 cm 5 cm 3m 3 ft 8. 3 in 4 in 9 ft 8 cm 6 cm 1 cm 10. An equilateral triangle with sides of 8 cm. 11. A square with sides of 18 inches. 12. A regular octagon with sides of 5 cm. © Milliken Publishing Company 9. 15 ft 10 cm 10 in 4 cm 2 cm 7. 13. A regular hexagon with sides of 10 ft. 14. A regular pentagon with sides of 6 yards. 15. A rhombus with sides of 9 mm. 24 MP4056 Area of Rectangles and Triangles The area of a figure is the number of square units needed to fill its interior. Area of some shapes can be found by using formulas. Area of Rectangle Area of Triangle 3 units 3 units 8 units 8 units Area = 24 square units OR Area = base x height 8 units x 3 units = 24 units2 Counting square units is more difficult than multiplication, but this triangle is really half of the rectangle to the left. 1 Area = 2 (base x height) Area = 12 (8 units x 3 units) = 1 (24 units2) 2 = 12 units2 Remember: Base and height must be perpendicular. Find the area of the figures below. Write the answers in the cross–number puzzle. ACROSS 12 1. 3. 4. 5. 7 13 5 16 16 14 10 8 8. 3 2 1 9x 8 4 x 5 6 perimeter = 60 7 8 DOWN 4 1. 2. 37 3. 13 5 4. 15 50 8 9 6 6. 3 3 7. 5 5 15 8 6 6 © Milliken Publishing Company 25 MP4056 Area of Trapezoids All trapezoids can be divided into a rectangle and one or two triangles. These areas can be found separately and then added together, or the formula for the area of a trapezoid can be used. Area Addition Method Formula Method 10 10 5 I 4 4 3 II 5 5 3 3 10 5 4 4 3 10 1 Rectangle = 10 x 4 = 40 1 ΔI = 2 (3 x 4) = 6 1 ΔII = 2 (3 x 4) = 6 total area = 52 units2 Area = 2 (height) (base 1 + base 2) 1 = 2 (4) (10 + 16) = 12 (4) (26) = 52 units2 Remember: Height is always perpendicular to the bases. Find the area of the following trapezoids. Follow the answers through the maze. 1. 2. 10 5 4 13 4. 7 5. 6 10 5 3 13 6. 14 13 2 8. 12 13 7. 7 5 4 3 5 3. 7 8 9 8. 10 4 10 8 9. 6 10 13 13 6 3 14 6 20 © Milliken Publishing Company 26 MP4056 10. 11. 12. 10 12 12 5 5 5 5 13 12 16 20 14. 13. 15. 12 3 5 15 12 7 12 15 3 5 9 © Milliken Publishing Company 27 MP4056 Area of Parallelograms A parallelogram can be divided into two congruent triangles and one rectangle. Then the area formulas can be applied: II 5 4 Notice what happens if we use the formula for parallelograms. 5 I 3 6 area of rectangle = 4 x 6 = 24 1 area of ΔI = 2 (3 x 4) = 6 1 area of Δ II = 2 (3 x 4) = 6 total area = 36 square units A = base x height A = 9x4 A = 36 square units Remember: Base and height must be perpendicular. Find the area of the parallelograms. Match answers with the words below to learn about Rear Admiral Grace Hopper. 1. 3 2. 7 3. 12 10 5 8 15 3 lady first COBOL 24 4. 5. 6. 7 10 2 4 5 1 computer 3 also of 7. 8. 9. 5 2 17 13 4 7 known © Milliken Publishing Company 5 3 coined 28 8 the MP4056 10. 11. 12. 3 9 3 8 10 5 4 10 5 phrase 13. was 14. 15 since 15. 18 8 5 12 12 3 2 she 13 bug inventors 16. 17. 18. 6 5 8 10 4 4 1 2 5 software ________ __________ 60 ________ 72 _________ 60 _____________________ _______________ 165 © Milliken Publishing Company 18 80 _________ 44 ________________. 180 ________________ 52 29 ________ 32 28 _________ 18 ______________________” _______ 216 ______ 20 28 ______________ 120 ______ 144 _______________ 40 ____________ ________________ 44 “___________ as one _________ 28 __________ 60 “______________ 240 18 ___________ 8 _________” 156 MP4056 Word Search As you find the words in the list below, shade in the squares containing the letters. When finished, the remaining letters (read horizontally from top to bottom) will explain what is so special about the number 1991 and the phrase, “some men interpret nine memos.” C O N C E N T R I C A I N S C R I B E D P E R A L S E C A N T T L C I N D T R R O M A G U E I S N A C N U N N M M B E E R O R W G A S E M I C I R C L E L C O S R D T H A T N A R I S U T H D E S T R I A M E I F O R R W T B A R D S D A N O D N E B A C K W A A R H E D I A M E T E R D S C Word List: ____ RADIUS DIAMETER CHORD ARC SEMICIRCLE CENTRAL ANGLE CENTER SECANT CONCENTRIC INSCRIBED CIRCUMSCRIBED TANGENT ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ . © Milliken Publishing Company 30 MP4056 Circumference Circumference is the distance around a circle. Think of circumference as the circle’s perimeter. Find circumference using the formula. Circumference = π (diameter) OR C = πd Remember: The diameter is twice the length of a radius. Find the circumference for the circle. Leave your answers in terms of π. Match your answers in the decoder to answer the riddle. 1. radius = 4 cm 5. radius = 11 ft 2. diameter = 18 m 6. diameter = 4 m 10. radius = 18 m 3. diameter = 10 cm 7. radius = 10 ft 11. radius = 9 ft 4. radius = 3 cm 8. diameter = 11 m 18π m A 4π m 11π m 3π in 10π cm 18π ft D E I K L 9. diameter = 3 in 8π cm 20π ft M 6π cm 36π m O P N 22π ft U What do mathematicians like to eat on Thanksgiving? ____ ____ ____ ____ ____ ____ ____ 10 5 ____ 2 © Milliken Publishing Company 1 10 ____ ____ 11 3 9 ____ ____ 7 10 9 ____ ____ ____ ____ . 2 1 31 4 6 8 MP4056 Area of a Circle The area of a circle is found by using the formula Area = π (radius)2 OR A = πr2 Remember: Use square units when finding area. Find the area of the circles described. Shade your answers below to reveal the name of the mathematician who invented the roulette wheel. 1. radius = 4 cm 11. diameter = 4 cm 2. diameter = 18 cm 12. radius = 10 cm 3. diameter = 10 m 13. radius =13 cm 4. radius = 7 in 14. diameter = 28 in 5. radius = 3 m 15. diameter = 40 cm 6. diameter = 12 ft 16. radius = 3 ft 7. radius = 12 in 17. diameter = 32 in 8. diameter = 30m 18. radius = 9 ft 9. diameter = 22 cm 19. radius = 18 cm 10. radius = 8 m © Milliken Publishing Company 20. diameter = 34 m 32 MP4056 Area of Irregular Shapes Dividing irregular shapes helps in finding their area. Try to form shapes that are listed below, then use the area formulas to solve for the total area. 1 1 Triangle: A = 2 bh Rectangle: A = bh Parallelogram: A = bh Trapezoid: A = 2 h(b1 + b2 ) Circle: A = πr 2 Use the decoder to find the animal with the highest blood pressure by solving for the area of each shape. 1. 2. 3. 5 5 5 10 10 10 8 10 12 10 3 2 10 10 2 8 15 4. 5. 6. 10 5 10 5 10 10 10 7. 13 13 6 12 8. 14 8 5 5 4 10 5 16 5 8 96 224 A E 9π 12 + 2 F ____ ____ ____ 5 © Milliken Publishing Company 8 7 136 88 G H 240 I 92 100 + 50π R T ____ ____ ____ ____ ____ ____ ____ 2 6 33 1 3 4 4 7 MP4056 Area of a Shaded Region To find the area of a shaded region, add or subtract areas of basic figures (rectangle, triangle, circle, and so on). Area of Large Circle = 72π = 49π Example: 5 2 2 — Area of Small Circle = 5 π = 25π Area of Shaded Ring = 49π – 25π = 24π units2 Find the area of the shaded regions below. Use your answers to reveal the game James Naismith invented. 1. 2. 3. 4 3 8 6 4 10 10 4. 5. 6. 10 2 10 7 10 6 12 6 7 3 5 4 4 8 10 7. 10 13 12 12 13 25π – 24 84 60 400 – 100π 33π 51 18 A B E K L S T ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 2 © Milliken Publishing Company 3 5 4 7 6 34 2 3 1 1 MP4056 Three–Dimensional Figures Unscramble the words below. All are terms related to three dimensional figures. Then place the circled letters in the blanks to reveal the diameter of a golf hole. 1. AEFC ____ ____ ____ ____ 2. OCNE ____ ____ ____ ____ 3. LOVEUM ____ ____ ____ ____ ____ ____ 4. HGIRT ____ ____ ____ ____ ____ 5. L N A ST EGIHTH ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 6. YAIPRDM ____ ____ ____ ____ ____ ____ ____ 7. TTADLIUE ____ ____ ____ ____ ____ ____ ____ ____ 8. BQEULIO ____ ____ ____ ____ ____ ____ ____ 9. LRTLAEA DGEE ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 10. ERXTVE ____ ____ ____ ____ ____ ____ 11. SDRAUI ____ ____ ____ ____ ____ ____ 12. YLCDREIN ____ ____ ____ ____ ____ ____ ____ ____ 13. UBCE ____ ____ ____ ____ 14. HEESPR ____ ____ ____ ____ ____ ____ 15. SPMRI ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ © Milliken Publishing Company 35 ____ ____ ____ ____ ____ ____ ____ MP4056 Surface Area of Right Prisms Right prisms have rectangles for lateral faces. Their bases can be any polygon. To find total surface area, add the area of all lateral faces and bases. 5 1 Example: area of base = 2 (3 x 4) = 6 4 3 area of lateral faces = (4 x 10) + (5 x 10) + (3 x 10) = 40 + 50 + 30 10 = 120 total area = 2 bases + lateral faces = 6 + 6 + 120 = 132 units2 Find the total area of the right prisms. 1. 2. 8 cm 5 cm 12 in 13 in 12 cm 20 in 5 in 4. 10 m 3. 4m 10 m 5m 10 mm 15 mm 3 mm 25 m 16 m 5. 6. 15 cm 12 in 5 in 4 in 10 in 25 cm 12 cm 6 in 9 cm 7. A cube with base edge of 5 cm. © Milliken Publishing Company 8. A right triangular prism whose bases are right triangles with sides 6 cm, 8 cm, and 10 cm, and whose height is 12 cm. 36 MP4056 Volume of Rectangular Prisms The volume of a rectangular prism is found by multiplying length, width, and height. Another way to think of it is finding the area of the base (length x width) and multiplying by height. Volume is measured in cubic units (units3). OR Volume = length x width x height Volume = (area of base) x height Find the missing measures in the chart below. length width height 1. 10 cm 5 cm 8 cm 2. 20 m 10 m 15 m 3. 14 in 6 in 9 in 4. 3 ft 4 ft 5 ft 5. 25 mm 10 mm 5 mm 6. 7 cm 8 cm 4 cm 7. 13 m 6m 3m 8. 24 ft 9 ft 16 ft 9. 8 in 12 in 15 in 10. 30 mm 50 mm 60 mm 11. 15 cm 8 cm 12. 10 m 13. 6 mm volume 480 cm3 15 m 750 m3 18 mm 2592 mm3 14. 100 ft 20 ft 100,000 ft3 15. 9m 12 m 1080 m3 © Milliken Publishing Company 37 MP4056 Volume of Right Prisms The volume of right prisms is found by multiplying the area of the base by the height of the prism. Remember: The bases are the parallel faces of the prism. Find the volume of the prisms described. Use your answers to reveal what Archimedes said upon discovering the principle of buoyancy in his bath. 1. 2. 12 m 4m 15 m 3m 10 m 10 m 5m 6m 8m 3. 4. 6 cm 25 cm 8m 12 cm 13 cm 13 cm 5 cm 18 cm 16 cm 12 cm 5. 5m 25 m 5m 10 m 14 m 10 m 6m 720 m3 3300 cm3 72 m3 E K A 1800 m3 R 540 cm3 U ____ ____ ____ ____ ____ ____ 3 © Milliken Publishing Company 4 5 38 3 1 2 MP4056 Surface Area of Regular Pyramids A regular pyramid has a regular polygon for its base and congruent isosceles triangles for its lateral faces. To find the total surface area, add the area of the base and the area of all lateral faces. Example: base edge = 5 cm lateral edge = 6.5 cm slant height = 6 cm The orange line is the height of the triangular face — also known as the slant height. 6.5 cm 6 cm 5 cm 5 cm Area of base = 5 x 5 = 25 cm2 1 Area of lateral face = 2 (6)(5) = 15 cm2 x 4 faces = 60 cm2 Total area = 25 + 60 = 85 cm2 Find the total surface area for the regular pyramids below. Use your answers to determine what the E represents in the formula E = mc2. 84 cm2 336 m2 112 cm2 756 m2 340 cm2 736 m2 N E G E Y R ____ ____ ____ ____ ____ ____ 4 3 6 m 10 13 m 5c 12 m 5. 6. m cm 15 m 17 5 cm 18 m © Milliken Publishing Company 16 m 3 12 m 8m m m 10 cm 15 m 2 3. 12 c 4c 6 cm 4. 5 2. cm 1. 1 8 cm 39 MP4056 Volume of Regular Pyramids The volume of any regular pyramid is found by finding the area of the base, multiplying it by the height (or altitude) of the pyramid, and then dividing the result by 3 (which is the same as multiplying by 13 ). Volume is measured in cubic units. Volume = 1 3 (area of base)(altitude of pyramid) The altitude of the pyramid is the distance from the center of the base to the tip (or vertex) of the pyramid. Notice the right triangle formed by the altitude, the slant height, and half the length of the base in a regular square pyramid: altitude slant height 1 base 2 Use this triangle and the Pythagorean Theorem to find needed measurements when they are not given. Example: Find the volume of a regular square pyramid with base edge of 6 cm and slant height of 5 cm. 32 + x2 = 52 Volume: 1 9 + x2 = 25 = 3 (area of base) (altitude) slant height 5 cm altitude x2 = 16 = 13 (6 x 6) (4) x = 4 = altitude = 1 (36) (4) 1 base 3 2 = 48 cm3 3 cm Find the volume of each of the following regular square pyramids using the information given. 1. base edge of 8 cm and altitude of 6 cm. 2. base edge of 10 cm and altitude of 3 cm. 3. base edge of 6 cm and altitude of 5 cm. 4. base edge of 5 cm and altitude of 3 cm. 5. base edge of 7 cm and altitude of 9 cm. 6. base edge of 8 cm and slant height of 5 cm. 7. base edge of 10 cm and slant height of 13 cm. 8. base edge of 24 cm and slant height of 15 cm. 9. slant height of 10 cm and altitude of 6 cm. 10. slant height of 17 cm and altitude of 15 cm. © Milliken Publishing Company 40 MP4056 Surface Area of Cylinders A cylinder’s surface area is made up of two circles (the top and bottom) and a rectangle (the lateral surface; picture the lateral surface as the label on a can). The rectangle’s dimensions include the circumference of the circle and the height of the cylinder. height circumferemce Total surface area: = 2 (area of circular base) + (circumference) (height) = 2πr2 + 2πrh r = 2πr (r+h) h Example: radius = 5 cm height = 12 cm Area of base = πr2 = π (5)2 = 25π Area of rectangle = 2πrh = 2π (5) (12) = 120π OR 2πr (r+h) = 2π (5) (5+12) = 2π (5) (17) = 170π cm2 2 bases = 50π cm2 + rectangle = 120π cm2 total = 170π cm2 Find the total surface area for the cylinders described below. Then use your answers to find the name of the wheel Blaise Pascal invented in a search for perpetual motion. 1. radius = 5 cm; height = 10 cm 2. radius = 4 cm; height = 10 cm 3. radius = 8 cm; height = 3 cm 4. radius = 10 cm; height = 7 cm 5. diameter = 24 m; height = 4 m 6. diameter = 30 m; height = 9 m 7. diameter = 10 m; height = 8 m 8. diameter = 18 m; height = 11 m 340π cm2 E 150π cm2 130π m2 H L ____ ____ ____ 8 1 176π cm2 384π m2 O 360π m2 R 112π cm2 T 720π m2 U W ____ ____ ____ ____ ____ ____ ____ ____ 4 5 3 2 7 4 8 8 4 ____ ____ ____ ____ ____ 6 © Milliken Publishing Company 1 4 41 4 7 MP4056 Surface Area of Cones A cone’s surface area is made up of one circular base and a triangular–shaped sector. Remember: A cone has a slant height, similar to that in pyramids. A CONE’S SURFACE: slant height slant height r r circle circumference circle area: A = πr2 triangular sector area A = 12 base x height A = 12 circumference x slant height 1 2 total surface area = πr + 2 (2πr) (slant height) A = 1 (2πr) (slant height) 2 OR = πr2 + πr (slant height) Find the total surface area of the cones described below. Use the decoder to find the answer to the question. 1. 2. 3. 4. m m 4 cm cm 5c m 12 7c 10 9 cm 3 cm 7m 5. 6. 7. 8. 30 m 15 12 m 20 m 8c cm 22 m m 18 cm 12 cm Who was the author of “Conic Sections,” a work containing propositions about curves that are created by slicing a cone with a plane? 24π cm2 A 84π cm2 44π cm2 171π cm2 I L N 286π m2 133π m2 525π m2 P S O 189π cm2 U ____ ____ ____ ____ ____ ____ ____ ____ ____ ____ 4 © Milliken Publishing Company 2 7 1 1 7 42 3 6 5 8 MP4056 Volume of Cylinders and Cones The volume of a cylinder is found by multiplying the area of the circular base by the height of the cylinder. V = (area of base) (height) 2 V = πr h The volume of a cone is found by taking 1 of the 3 product of the area of the base and the height (altitude) 1 V = 3 (area of base) (height) V = 13 πr 2h h r radius h Recall the right triangle created by the radius, altitude, and slant height. altitude slant height radius Use the Pythagorean Theorem to find the needed measurements when they are not given. Find the volume of the figure described. Follow your answers through the maze. 1. cylinder with radius 8 m; height 12 m 2. cone with radius 6 m; altitude 10 m 3. cylinder with radius 5 m; height 10 m 4. cone with radius 12 cm; altitude 16 cm 5. cylinder with radius 18 cm; height 20 cm 6. cone with radius 9 m; altitude 7 m 7. cylinder with radius 3 cm; height 2 cm 8. cone with radius 5 m; altitude 12 m 9. cylinder with radius 11 cm; height 7 cm 10. cone with radius 15 cm; altitude 10 cm © Milliken Publishing Company 11. cylinder with diameter 10 m; height 8 m 12. cylinder with diameter 20 m; height 20 m 13. cylinder with diameter 18 m; height 5 m 14. cone with diameter 6 cm; altitude 4 cm 15. cone with diameter 10 cm; altitude 9 cm 16. cone with diameter 16 m; altitude 6 m 17. cylinder with diameter 4 m; height 2 m 18. cone with radius 4 cm; slant height 5 cm 19. cylinder with diameter 8 m; height 12 m 20. cone with radius 12 cm; slant height 13 cm 43 MP4056 Crossword Puzzle Across Down 2 2 2 1. longest side of a right triangle 2. a + b = c 3. polygon with equal sides 4. four–sided polygon 6. larger than 90° 5. parallelogram with four equal sides 8. base of a cone 7. no equal sides 9. quadrilateral with two pairs of parallel sides 8. lateral face is a rectangle, bases 12. angles adding to 180° are circles 13. ratio of opposite side to hypotenuse 10. smaller than 90° 14. ratio of opposite side to adjacent side 11. equal 15. polygon with equal angles 16. quadrilateral with four right angles 19. distance around a circle 17. six–sided polygon 21. nonparallel, nonintersecting lines 18. lines that form right angles 22. ten–sided polygon 20. 90° angle 23. half of a diameter Word List acute circle circumference congruent cylinder decagon equiangular equilateral hexagon hypotenuse obtuse parallelogram perpendicular Pythagorean quadrilateral radius rectangle rhombus right scalene sine skew supplementary tangent 1 2 3 4 5 6 8 7 10 9 11 12 13 14 15 16 18 17 19 20 21 22 23 © Milliken Publishing Company 44 MP4056 Answer Key Page 1: ONE Page 2: EUCLID’S ELEMENTS Page 3: 1. 2. 3. 4. 5. Page 4: All in Philadelphia Page 6: congruent: GOTTFRIED LEIBNIZ Page 7: The acute angles of a right triangle are complementary Page 8: 1. x = 98° , y = 82° 2. x = 75° , y = 135° z = 105° 3. x = 160° , y = 50° Page 9: CARL FRIEDRICH GAUSS Page 12: THE TRIPLE JUMP Page 13: Page 14: right obtuse acute acute straight 1a. b. c. d. e. f. g. ΔPRS RP RS ∠P LN ∠M ∠S 6. 7. 8. 9. 10. obtuse straight obtuse right acute 11. 12. 13. 14. 15. Page 5: straight right obtuse acute straight 16. 17. 18. 19. 20. NIKOLAI LOBACHEVSKY supplementary: ISAAC NEWTON 4. x = 30° , y = 60° 5. w = 110° , x = 150° y = 100° , z = 70° 2a. b. c. d. e. f. g. obtuse obtuse acute obtuse straight 6. w = 85° , x = 45° y = 90° , z = 50° 7. x = 115° , y = 65° z = 75° Page 11: PYTHAGOREAN TRIPLES NDBLI IL ND LB BD NI ∠L JAMES ABRAM GARFIELD 3a. b. c. d. e. f. g. QARP PR DA ∠P TD RT ∠QAR h. i. j. k. ∠PRA TRAD PQAR RTDA Page 15: THE METHOD OF EQUILIBRIUM Answer Key Page 16: Page 17: S M E R S E O O I O E M U C U E F D N D Y W H A O U N H D S P E R P E N D I C U L A R I G R P R J M R T X Q D Y E J E H E E A E A V D R P A S A S H W U P V T H G W G K T R T K O O O L T T T H G I R E U W G A K G H H N T P T S U R S E S O L Z S C A L E N E Y U E S T C I G A K U B L A D J A C E N T P A S M A K E T N I L A C I T R E V N E I O N N A E I V S E A G A S S O E A S D M M T B M S R L W H I G I H N O R S R F U W A E N T K M S P A D B G U T S S H B A S E R N A V U R A H M L J K E Q C G I V E L J F U R N E S A I U O K D B M E A A E D Y V L S B K D U E N L L C E D R L S M R K A E T E V C B A K Y E A L H R X E A B I O T L L S U P P L E M E N Y A R Y S U N E N X E T R E V I C R V X A S N C D A ALASKA, CALIFORNIA, HAWAII, and WASHINGTON .... .... .... .... 2 .... ....3 8 0 1 .... 2 .... 2 .... 3 .... .... .... .... .... .... .... 4 .... .... 1 4 5 2 .... ............................ .... .... .... .... 5 6 7 .... .... .... 3 1 8 0 5 ........ .... .... .... .... .... .... .... .... .... 8 .... .... .... .... 0 1 1 0 ........ .................... ........ .... 9 10 .... .... 1 3 5 .... 2 3 .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... 5 0 0 .... .... ........ ........ .... 1 Page 18: Page 19: 1a. b. c. d. e. f. g. h. i. 24 10 26 26 13 13 90° 40° 100° 2a. b. c. d. e. f. g. h. i. j. 6 60° 120° 90° 60° 60° 30° 30° 3 equilateral 3a. b. c. d. e. f. g. h. 16 16 2 16 45° 45° 90° 90° 45° Answer Key Page 20: 1. x = 14 2. y = 16 3. z = 10 , a = 4 Page 21: The Golden Ratio was used to design the pyramids and also Greek buildings and artifacts. Page 22: 1. x = 8, y = 8 2 2. a = 12, b = 7.5 3. c = 16, d = 8 e = 16 Page 23: Mary Dixon Kies received a patent for her weaving machine in 1809. Page 24: 206 Page 25: 4. b = 6 , c = 3 , d = 5 5. e = 13 , f = 15, g = 10 4. f = 13.5, g = 13.5 h = 22.5, i = 18 5. r = 8, s = 8, t = 12 u = 12 19 4 2 5 3 3 6 5 3 7 6 88 1 0 6. v = 36, w = 15 0 4 0 2 6 area = 46 area = 40 area = 50 area = 54 area = 138 2 8 6. j = 5 , k = 14 7. m = 15 6 4 3 4 6. 7. 8. 9. 10. 1 Page 26–27: 1. 2. 3. 4. 5. area = 72 area = 180 area = 72 area = 36 area = 48 11. 12. 13. 14. 15. area = 174 area = 52 area = 42 area = 252 area = 27 Page 29: She was known as the “first lady of software” since she was one of the inventors of COBOL. She also coined the phrase “computer bug.” Page 30: A palindrome is a number or word that is the same forwards and backwards. Page 31: PUMPKIN PI A LA MODE Page 32: PASCAL Page 33: THE GIRAFFE Page 34: BASKETBALL Page 35: FOUR AND A QUARTER INCHES Answer Key 4. 450 mm2 5. 1008 cm2 6. 352 in2 Page 36 1. 392 cm2 2. 660 in2 3. 1004 m2 Page 37: 1. 2. 3. 4. 5. Page 38: EUREKA Page 40: 1. 128 cm3 2. 100 cm3 3. 60 cm3 Page 41: THE ROULETTE WHEEL Page 43: 1. 2. 3. 4. 5. Page 44: Across © Milliken Publishing Company 400 cm3 3000 m3 756 in3 60 ft3 1250 mm3 6. 7. 8. 9. 10. 7. 150 cm2 8. 336 cm2 224 cm3 234 m3 3456 ft3 1440 in3 90,000 mm3 11. 12. 13. 14. 15. Page 39: ENERGY 4. 25 cm3 5. 147 cm3 6. 64 cm3 768π m3 120π m3 250π m3 768π cm3 6480π cm3 2. 3. 6. 8. 9. 12. 13. 14. 15. 19. 21. 22. 23. 6. 7. 8. 9. 10. 7. 8. 9. 10. Page 42: 189π m3 18π cm3 100π m3 847π cm3 750π cm3 Pythagorean equilateral obtuse circle parallelogram supplementary sine tangent equiangular circumference skew decagon radius 48 4 cm 5m 24 mm 50 ft 10 m 11. 12. 13. 14. 15. APOLLONIUS 200π m3 2000π m3 405π m3 12π cm3 75π cm3 Down 40 cm3 1728 cm3 512 cm3 1280 cm3 1. 4. 5. 7. 8. 10. 11. 16. 17. 18. 20. 16. 17. 18. 19. 20. 128π m3 8π m3 16π cm3 192π m3 240π cm3 hypotenuse quadrilateral rhombus scalene cylinder acute congruent rectangle hexagon perpendicular right MP4056