Unit 3 Similar Polygon Practice. Contents

Transcription

1 Unit 3 Similar Polygon Practice Contents 1) Similar Polygon Practice ) Intro to Similar Triangles ) Dilations He Said, She Said ) Similar Polygons ) Valentine s Day Couples ) Similar Triangles ) Working with Similar Triangles ) More Similar Triangles ) Ways to Prove Triangles Similar ) CSI Similarity and Proportions ) Drawing Triangle Families ) Scrambled Proofs ) Algebra and Similarity ) Similar Polygon Meaningful Notes ) Similarity Transformations ) Dilations and Similarity Practice

2 Unit 3 Similar Polygon Practice 2 Similar Polygon Practice Assume ΔABC ~ ΔDEF. Determine which theorem you can use to prove each set of triangles similar. Solve for x.

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6 Unit 3 Intro to Similar Triangles Intro to Similar Triangles This is the symbol for. Similar polygons are the same but not necessarily the same. Similar polygons have these features angles sides To see which angles are congruent, match up corresponding angles. Try these by filling in the missing angles. The scale factor tells you what to multiply one polygon s length by in order to get the second polygon s length. The triangles are similar. What is the scale factor? ΔABC ~ ΔDEF, AB = 10 m, AC = 15 m, BC = 14 m The scale factor is 0.5. Find the missing lengths. 6

7 Unit 3 Intro to Similar Triangles Intro to Similar Triangles (continued) 7

8 Unit 3 He Said She Said Dilations He Said, She Said Instructions: Each of the 10 cards will have statements from two students. Read carefully, and choose the name of the student who made the correct statement. Then, justify your answer in the space provided. Note: Assume all dilations have the origin as the center of dilation. 1) is correct because 2) is correct because 8

9 Unit 3 He Said She Said 3) is correct because 4) is correct because 9

10 Unit 3 He Said She Said 5) is correct because 6) is correct because 10

11 Unit 3 He Said She Said 7) is correct because 8) is correct because 11

12 Unit 3 He Said She Said 9) is correct because 10) is correct because 12

13 Unit 3 Similar Polygons Similar Polygons 1) If polygons are similar then what do you know about the corresponding sides and the corresponding angles? Given the similar figures, name all pairs of corresponding sides and angles. Look at the similarity statement to help. 2) PQR ~ DEF 3) LMN ~ RST 4) ABCD ~ HGFE QP Q LM L AB A PR P MN M BC B RQ R NL N CD C DA D Use the similar polygons above to write the statement of proportionality for each: = = = = = = = Complete the similarity statement for the similar figures and then find the scale factor. Reduce fractions! 5) 6) 7) LKM ~ CBAD ~ RSPQ ~ Scale Factor: Scale Factor: Scale Factor: 8) 9) 10) HJG ~ NPM ~ KJML ~ Scale Factor: Scale Factor: Scale Factor: 13

14 Unit 3 Similar Polygons The two polygons are similar. Write a proportion and solve for x. 11) 12) 13) Complete the similarity statement for the similar figures and then find the scale factor. Next, write proportions and SOLVE for the missing lengths. 14) 15) 16) 17) 14

15 Unit 3 Similar Polygons Are these triangles similar by the AA~ Postulate? Answer yes or no. If the triangles are similar, write a similarity statement. 18) 19) Similar: YES NO Similar: YES NO ADE ~ EDF ~ Find the angle measurements and set up proportions to find all missing side lengths. Notice the triangles are similar by AA~. 20) Flipped OR Twisted?? 21) Flipped OR Twisted?? m 1 = m A = m D = m A = m TBM = m T = Proportion to find x: Proportion to find x: Proportion to find y: Proportion to find y: Given two similar figures, find the scale factor and the ratio of the perimeters from the SMALL to the BIG. 22) 23) Scale Factor: Ratio of Perimeters: Scale Factor: Ratio of Perimeters: 15

16 Unit 3 Valentine s Day Couples Valentine s Day Couples Find the value that relates to similar triangles to discover these famous couples. Show your thinking. Each independent question pertains to the diagram at the right. 1) AB = 20, DB = 15, BC = 16, BE = 2) DB = 12, BE = 8, EC = 4, AD = 3) AB = 84, DB = 63, EC = 16, BE = 4) AB = 24, DB = 8, DE = 6, AC = 5) DB = 15, AD = 9, AC = 18, DE = 6) AB = 18, AD = 3, BE = 16, EC = 7) AD = 2, DB = 8, BC = 12, BE = 8) AD = 2, DB = 8, AC = 9, DE = 9) AD = 5, DB = 15, BE = 9, EC = 10) AD = 8, DB = 16, DE = 21, AC = 11) AD = 15, DB = 20, BE = 12, EC = 12) AD = 5, DB = 12.5, EC = 4, BE = 13) DB = 25, BC = 32, BE = 20, AD = 14) AD = 12, DB = 24, EC = 11, BE = 15) AB = 40, AD = 8, AC = 25, DE = 16) BE = 8, EC = 4, DE = x, AC = x = 4, x = 16

17 Unit 3 Kuta Similar Triangles Similar Triangles State if the triangles in each pair are similar. If so, state how you know they are similar and complete the similarity statement. 1) ΔUTS ~ 2) ΔCBA ~ 3) ΔVUT ~ 4) ΔJKL ~ 5) ΔSTU ~ 6) ΔJKL ~ 17

18 Unit 3 Kuta Similar Triangles 7) ΔTUV ~ 8) ΔTUV ~ 9) ΔHGF ~ 10) ΔFGH ~ 11) ΔFED ~ 12) ΔEFG ~ 18

19 Find the missing length. The triangles are similar. Unit 3 Kuta Similar Triangles 13) 14) 15) 16) Solve for x. The triangles in each pair are similar. 17) 18) 19

20 Working with Similar Triangles Unit 3 Working with Similar Triangles Find the labeled lengths 20

21 More Similar Triangles Unit 3 More Similar Triangles Find the area of the following triangles. Area = ½ bh 5. What is the ratio of the sides in #1 and #2? 6. What is the ratio of the sides in #3 and #4? 7. What is the ratio of the areas in #1 and #2? 8. What is the ratio of the areas in #3 and #4? 9. What can you conclude about this? Find the ratio of the areas in the following sets of similar triangles with corresponding sides labeled. 21

22 Unit 3 Ways to Prove Triangles Similar Ways to Prove Triangles Similar Identify which property will prove these triangles similar. 22

23 CSI Similarity and Proportions Unit 3 CSI Similarity and Proportions 23

24 Unit 3 CSI Similarity and Proportions 24

25 Unit 3 CSI Similarity and Proportions 25

26 Unit 3 CSI Similarity and Proportions 26

27 Unit 3 CSI Similarity and Proportions 27

28 Drawing Triangle Families Part 1: Drawing Triangle Families Unit 3 Drawing Triangle Families Remember: A Triangle Family is a sequence of triangles where the ratio of the sides are always the same. Task #1: The first two members of the 1 2 triangle family are drawn below. Draw in the 3rd and 4 th members of this family in the grid below. Task #2: Explain why this triangle family is called the 1 triangle family. Your answer should have the word ratio and 2 reduce in it. My Answer: Task #3: Draw the first three members of the 2 Triangle Family 3 Task #4: Mr Schneider has drawn 3 random triangles below and he wants to create a family out of these triangles. He s put YOU in charge of finding these other triangles. Your Job: For each triangle below, draw 2 more triangles that are part of the same family. The drawings do not have to be to scale just make sure the numbers are correct. Remember: The ratios should be the same in every triangle! 28

29 Unit 3 Drawing Triangle Families Part 2: Triangle Families & Missing Measurements Task #1: Mr Schneider has drawn all of the triangles below so that they are all part of the same family, but he got lazy and forgot to fill in the missing measurements. Use triangle families and the Pythagorean Theorem to find all missing measurements. You may use a calculator Possible Answers Task #2: Mr Schneider has drawn all of the triangles below so that they are all part of the same family, but he got lazy and forgot to fill in the missing measurements. Use triangle families and the Pythagorean Theorem to find all missing measurements Possible Answers Task #3: Mr Schneider has drawn all of the triangles below so that they are all part of the same family, but he got lazy and forgot to fill in the missing measurements. Use triangle families and the Pythagorean Theorem to find all missing measurements Possible Answers 29

30 Part 3: Playing Pool with Triangles Unit 3 Drawing Triangle Families Fact: A good pool player knows their. Whenever you shoot a pool ball against a wall, it bounces off at the same angle it came in, creating a triangle. As the ball continues to bounce, it creates a family of triangles. In the diagram above, three triangles are created. If we write the sides of these triangles as ratios, we get: Notice: These are all part of the 1 triangle family 2 1 2, 5 10, 2 4 Task #1: Draw a line in the diagram above showing where the pool ball will go next Check Yourself: You should have created a new triangle in the diagram above. Write the sides of this triangle as a ratio: This ratio should reduce to 1. If it doesn t, check with your group or ask Mr Schneider! 2 Task #2: Use the diagram below to draw the next 4 places where the ball will bounce off of the walls What triangle family did you draw in the diagram above? My Answer: Fun Fact: This is how lasers work too! 30

31 Unit 3 Drawing Triangle Families Part 4: Pool Without the Grid Task #1: Use your knowledge of triangle families and the Pythagorean Theorem to determine each missing measurement after Mr Schneider shoots his pool ball: FC = HJ = JG = GD = DC = AB = BC = KJ = JD = LE = AG = BF = CF = DA = DL = LJ = BC = 31

32 Scrambled Proofs Unit 3 Scrambled Proofs The REASONS to the following proofs are scrambled. Find the REASON that corresponds to each STATEMENT. Place the STATEMENT number in front of its corresponding REASON. 1) Given: ΔABC with right ACB CD is altitude to AB Prove: (CB) 2 = BD AB Statements 1. ΔABC with right ACB; CD is altitude to AB 2. CD AB Reasons All Right angles are congruent AA~: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar 3. CDB is a right angle Given 4. ACB CDB 5. B B 6. ACB~ CDB Corresponding sides of similar triangles are in proportion In a proportion, the product of the means equals the product of the extremes An altitude of a triangle is a segment from any vertex o the triangle perpendicular,, to the line containing the opposite side. 7. BD = CB CB AB Perpendicular lines form right angles 8. (CB) 2 = BD AB Reflexive Property of Congruence 32

33 Unit 3 Scrambled Proofs 2) Given: Parallelogram ABCD Prove: FE BE = AE FC Statements Reasons 1) Parallelogram ABCD Congruent segments have equal measure 2) AC BC Corresponding sides of similar triangles are in proportion 3) D BCE Given 4) E E 5) ΔADE ~ ΔFCE In a proportion, the product of the means equals the product of the extremes A parallelogram has two sets of opposite sides congruent 6) FE = FC AE AD Reflexive Property of Congruency 7) AD BC 8) AD = BC 9) FE = FC AE BC A parallelogram has two sets of opposite sides parallel AA ~: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar If two parallel lines are cut by a transversal, then the corresponding angles are congruent 10) FE BE = AE FC Substitution Property 33

34 Unit 3 Algebra and Similar Triangles Algebra and Similarity Directions: As you find the missing values in these diagrams, cross out the answer in the answer box at the end. When finished, add the remaining values in the table. Be sure to show your work. 1) x = 2) y = 3) x = 4) x = Assume the ratio of similitude of the triangles is not 1:2 5) x = 6) x = Assume these similar pentagons are drawn with their corresponding sides in the same positions. 34

35 7) x = 8) x = The ratio of similitude is 1:3 Unit 3 Algebra and Similar Triangles 9) x = ΔBAC ~ ΔACD 10) x = The sum of the remaining values in the Answer Box is 35

36 Identify Similar Figures Similar Polygon Meaningful Notes Unit 3 Similar Polygon Meaningful Notes Similar Polygons o Have the but are different in o Two polygons are similar if their vertices are paired so that corresponding corresponding their lengths have the same ratio the ~ means Naming Similar Polygons The order of vertices is critically important The order identifies angles and sides To properly name a polygon o Choose a vertex in one polygon, then choose the corresponding vertex in the other polygon. Move in the same direction around both polygons. Writing a similarity statement A similarity statement is something that o o o Tells which figures are Identifies the Identifies the Write a similarity statement for the similar trapezoids above. 36

37 Unit 3 Similar Polygon Meaningful Notes Side Side Side (SSS) ~ If the sides of two triangles are in proportion, then the triangles are similar. Side Angle Side (SAS) ~ If an angle of one triangle is congruent to an angle of another triangle and the sides including those angles are in proportion, then the triangles are similar. Angle Angle (AA) ~ If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Scale Factors Scale factors are comparison of the lengths of the corresponding sides of one figure to another expressed as a numerical ratio How a scale factor is written depends on the order of comparison 37

38 Unit 3 Similar Polygon Meaningful Notes For each pair of figures list the corresponding congruent angles, the proportional sides, and write a similarity statement. 1) a) Corresponding congruent angles b) proportional sides c) similarity statement 2) a) Corresponding congruent angles b) proportional sides c) similarity statement Solving for missing measures and Scale Factors Scale factors are comparison of the lengths of the corresponding sides of one figure to another expressed as a numerical ratio o How a scale factor is written depends on the order of comparison (big to small or vice versus) o A scale factor can be used in your proportions in the place of another ratio to make solving easier 38

39 For each pair of similar figures, find the value of the variables(s). Show your thinking. Unit 3 Similar Polygon Meaningful Notes 1) x = 2) x = 3) x = y = 4) x = 39

40 Unit 3 Similar Polygon Meaningful Notes Part One: For each pair of similar figures list a) the corresponding congruent angles, b) the proportional sides and c) write a similarity statement 1) a) Corresponding congruent angles b) proportional sides c) similarity statement 2) a) Corresponding congruent angles b) proportional sides c) similarity statement Part Two: Determine if each pair of figures is similar and state why or why not. Show your thinking. 3) Similar YES or NO 4) Similar YES or NO 5) Similar YES or NO 6) Similar YES or NO 40

41 Unit 3 Similar Polygon Meaningful Notes Part Three: For each pair of similar figures find the value of the variable(s). Show all of your work. 7) x = 8) x = 9) x = 10) x = 11) x = 12) x = 13) x = 14) x = 41

42 Determine if the figures are similar. Explain. Similarity Transformations Unit 3 Similarity Transformations 1) 2) Triangle PQR and triangle STU 3) JKLMN and JPQRS 42

43 Unit 3 Similarity Transformations Find the sequence of similarity transformations that map one figure to another. Write the coordinate notations for each transformation. 4) 5) Transformations: Transformations: Coordinate Notation: Coordinate Notations: 6) Transformations: Coordinate Notations: 43

44 Unit 3 Dilations and Similarity Practice Dilations and Similarity Practice Determine if the transformation is a dilation. 1) 2) 3) Determine the center of dilation and the scale factor. Center of dilation: Scale Factor: 44

45 Unit 3 Dilations and Similarity Practice Tell whether one figure appears to be a dilation of the other figure. Explain. 1) 2) 3) Square A is a dilation of square B. What is the scale factor? Determine if the transformation of figure A to figure B on the coordinate plane is a dilation. Verify ratios of corresponding side lengths for a dilation. 4) 5) 45

46 Determine the center of dilation and the scale factor of the dilation. Unit 3 Dilations and Similarity Practice 6) Scale Factor 7) Scale Factor 46