Honors Geometry Companion Book, Volume 2

This companion book to Thinkwell's Geometry course includes: • Illustrated lesson notes • Lesson worksheets • Practice exercises • Review worksheets • Suggested pacing guide • Formula and resource worksheets Geometry Volume 2 covers: Ch 7 - Similarity Ch 8 - Right Triangles and Trigonometry Ch 9 - Extending Perimeter, Circumference and Area Ch 10 - Spatial Reasoning Ch 11 - Circles Ch 12 - Transformational Geometry

A companion book for Thinkwell's Honors Geometry online video course. Volume #2 covers Chapters 7 ‒ 12.

No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without prior written permission of Thinkwell Corporation unless such copying is expressly permitted by federal copyright law. Address inquiries to: Permissions, Thinkwell Corp., 505 E. Huntland Drive, Suite 150, Austin, TX 78752.

Before we get too far into the fun, I wanted to personally introduce myself: I'm your virtual instructor, Professor Burger. Welcome to Thinkwell's wonderful world of Honors Geometry!

Together, you and I will explore the ideas that are foundational to Geometry. I hope you ’ ll let me help you in MAKING MEANING of the math ideas we ’ ll see together — there ’ s no need to memorize if you focus on deep understanding. I invite (urge) you to take the time to truly think through the math we ’ ll explore together and to mindfully practice the skills behind the ideas every day. If you do, you ’ ll not only succeed in Honors Geometry, but you ’ ll also be on solid ground for all the other math in your future! About This Book This is a companion book to Thinkwell's Honors Geometry online video course. Use this book as a complement to the online materials. I always say that to learn math you must DO MATH! I encourage you

to put pencil to paper and • take notes in this book

• highlight key concepts and earmark ideas you want to remember • doodle, sketch, and visualize the math ideas presented in each topic.

In a nutshell, make this book your own and keep it by your side as you study the concepts in this course. This book is divided into twelve chapters. Within each chapter are a series of Geometry topics. Every topic online contains my Video Lessons along with an electronic version of the Notes, Practice questions and Worksheet questions (although I wish we ’ d call them “ Funsheets ” , but that ’ s another story). How To Use This Book Use this book alongside the online course and TAKE NOTES here while watching my video lessons. Your own notes are a key to your own success — I promise. Since this book summarizes the concepts, vocabulary, and key examples presented in the Video Lessons, it is a great tool to help you navigate the videos — but this companion book is not intended as a shortcut to replace the Video Lessons. To get the most out of this learning experience, I urge you to watch (and think through) all of the online Video Lessons. Maybe even watch some twice! The online Geometry course offers lots of opportunities to practice the skills you ’ ll need for success in Geometry. Each topic ’ s Worksheet Practice and Interactive Practice is a collection of questions connected to the content presented in the Video Lessons. I've included those questions here in this book, so you can explore them offline and spend time really thinking through each question. I always say that the best way to learn math is to DO MATH. So, take advantage of all the opportunities to practice what you've learned!

Finally, at the end of each section, I've included a Review Worksheet to give you even more opportunity to review and practice the concepts

you learned in the Video Lessons. Put pencil to paper (or pen, if you dare) to answer each question. I recommend you complete these Reviews before taking the course Quizzes and Tests online. LET'S GO! I look forward to our Honors Geometry journey together! Remember to make meaning and focus on deep understanding … and also remember that YOU CAN DO IT! Have fun! If you have any questions, please reach out to my friends at Thinkwell. Email them at support@thinkwell.com. Also, I ’ m on Twitter @ebb663, if you want to say, “ hello ” .

I wish you all the best in your Geometry success!

Your virtual teacher,

— Prof. B.

vii

TABLE OF CONTENTS

Tips for Success�� 1 Suggested Pacing Guide��3 Chapter 7: Similarity��5 7.1 Similarity Relationships��7 7.1.1 Ratio and Proportion ��9 7.1.2 Ratios in Similar Polygons ��15 7.1.3 Triangle Similarity: AA, SSS, and SAS �� 19 7.1 Review Worksheet ��25 7.2 Applying Similarity �� 29 7.2.1 Applying Properties of Similar Triangles �� 31 7.2.2 Using Proportional Relationships ��37 7.2.3 Dilations and Similarity in the Coordinate Plane ��42 7.2 Review Worksheet ��48 Chapter 8: Right Triangles and Trigonometry��53 8.1 Trigonometric Ratios��55 8.1.1 Similarity in Right Triangles �� 57 8.1.2 Trigonometric Ratios ��62 8.1.3 Solving Right Triangles ��68 8.1 Review Worksheet ��74 8.2 Applying Trigonometric Ratios �� 79 8.2.1 Angles of Elevation and Depression �� 81 8.2.2 Law of Sines and Law of Cosines ��85 8.2.3 Vectors ��90 8.2 Review Worksheet ��96 Chapter 9: Extending Perimeter, Circumference, and Area�� 101 9.1 Developing Geometric Formulas ��103 9.1.1 Developing Formulas for Triangles and Quadrilaterals �� 105 9.1.2 Developing Formulas for Circles and Regular Polygons ��114 9.1.3 Composite Figures ��119 9.1 Review Worksheet ��124 9.2 Applying Geometric Formulas �� 129 9.2.1 Perimeter and Area in the Coordinate Plane �� 131 9.2.2 Effects of Changing Dimensions Proportionally �� 136 9.2.3 Geometric Probability ��142 9.2 Review Worksheet ��149

TABLE OF CONTENTS

Chapter 10: Spatial Reasoning��153 10.1 Three-Dimensional Figures��155 10.1.1 Solid Geometry ��157 10.1.2 Representations of Three-Dimensional Figures ��162 10.1.3 Formulas in Three Dimensions ��168 10.1 Review Worksheet ��175 10.2 Surface Area and Volume ��179 10.2.1 Surface Area of Prisms and Cylinders ��181 10.2.2 Surface Area of Pyramids and Cones ��189 10.2.3 Volume of Prisms and Cylinders ��197 10.2.4 Volume of Pyramids and Cones ��204 10.2.5 Spheres �� 211 10.2 Review Worksheet ��218 Chapter 11: Circles��227 11.1 Lines and Arcs in Circles��229 11.1.1 Lines That Intersect Circles ��231 11.1.2 Arcs and Chords ��237 11.1.3 Sector Area and Arc Length ��243 11.1 Review Worksheet ��247 11.2 Angles and Segments in Circles ��251 11.2.1 Inscribed Angles ��253 11.2.2 Angle Relationships in Circles ��260 11.2.3 Segment Relationships in Circles ��267 11.2.4 Circles in the Coordinate Plane��272 11.2 Review Worksheet ��277 Chapter 12: Transformational Geometry��283 12.1 Congruence Transformations��285 12.1.1 Refl ections ��287 12.1.2 Translations ��293 12.1.3 Rotations ��298 12.1.4 Compositions of Transformations��304 12.1 Review Worksheet ��309 12.2 Patterns ��315 12.2.1 Symmetry ��317 12.2.2 Tessellations ��323 12.2.3 Dilations ��330 12.2 Review Worksheet ��335 Blank Graph Paper�� 339 Formulas & Symbols��345

Use this book to help you stay organized.

Check out the suggested pacing guide in this book or download the online Lesson Plan and create a study schedule for yourself. Your schedule will be your plan for Geometry success!

Be an active learner. Before you begin studying, collect the tools you'll need: a pencil, scratch paper, highlighters, or graph paper are great things to have on-hand.

As you watch the Video Lessons online, work out the examples along with Prof. Burger on the Lesson Notes here (or on your own paper). Highlight important points in the Lesson Notes, and earmark topics you want to go back to review before a Quiz or Test.

Practice as you go. After each Video example, complete the Worksheet questions for that example. Once you've watched all the video lessons and answered all the Worksheet questions, check your understanding by completing the Practice question set. Go online to check your answers and to see answer feedback with step- by-step explanations. Review to remember.

Before a Quiz or Test, complete the Review Worksheet and re-do any exercises you need extra practice to master.

Reach out if you need help! Have questions? Need help? Reach out to us at support@thinkwell.com. We're here to help!

Suggested Pacing guide

This pacing guide follows a 36-week plan to sequentially progress through Thinkwell's Honors Geometry online course. Since the course is self-paced, feel free to go as quickly or as slowly through the material as you need to – this guide is just a suggestion. The list below corresponds with Thinkwell's Honors Geometry online course scope and sequence. 口 WEEK 1: – 1.1 Points, Lines, Planes, and Angles

口 WEEK 2: – 1.1 Points, Lines, Planes, and Angles (Cont.) 口 WEEK 3: – 1.2 Coordinate and Transformation Tools 口 WEEK 4: – 1.1-1.2 Online Assessment 口 WEEK 5: – 2.1 Inductive and Deductive Reasoning 口 WEEK 6: – 2.2 Mathematical Proof 口 WEEK 7: – 2.1-2.2 Online Assessment 口 WEEK 8: – 3.1 Lines with Transversals 口 WEEK 9: – 3.2 Slope and the Equation of a Line & 3.1-3.2 Online Assessment

口 WEEK 10: – 4.1 Triangles and Congruence 口 WEEK 11: – 4.2 Proving Triangle Congruence 口 WEEK 12: – 4.1-4.2 Online Assessment 口 WEEK 13: – 5.1 Segments in Triangles 口 WEEK 14: – 5.2 Relationships in Triangles 口 WEEK 15: – 5.1-5.2 Online Assessment 口 WEEK 16: – 6.1 Polygons and Parallelograms

口 WEEK 17: – 6.2 Other Special Quadrilaterals 口 WEEK 18: – 6.1-6.2 Online Assessment and Modules 1-17 Online Assessment

口 WEEK 19: – 7.1 Similarity Relationships 口 WEEK 20: – 7.2 Applying Similarity 口 WEEK 21: – 7.1-7.2 Online Assessment 口 WEEK 22: – 8.1 Trigonometric Ratios 口 WEEK 23: – 8.2 Applying Trigonometric Ratios 口 WEEK 24: – 8.1-8.2 Online Assessment

口 WEEK 25: – 9.1 Developing Geometric Formulas 口 WEEK 26: – 9.2 Applying Geometric Formulas 口 WEEK 27: – 9.1-9.2 Online Assessment 口 WEEK 28: – 10.1 Three-Dimensional Figures & 10.2 Surface Area and Volume 口 WEEK 29: – 10.2 Surface Area and Volume 口 WEEK 30: – 10.1-10.2 Online Assessment 口 WEEK 31: – 11.1 Lines and Arcs in Circles 口 WEEK 32: – 11.2 Angles and Segments in Circles 口 WEEK 33: – 11.1-11.2 Online Assessment 口 WEEK 34: – 12.1 Congruence Transformations 口 WEEK 35: – 12.2 Patterns & 12.1-12.2 Online Assessment 口 WEEK 36: – Modules 19-35 Online Assessment

Chapter 7 Similarity

7.1

Similarity Relationships

7.1.1 Ratio and Proportion

Key Objectives • Write and simplify ratios. • Use proportions to solve problems. Key Terms • A ratio compares two numbers by division. • A proportion is an equation stating that two ratios are equal. • In the proportion =

a b c d , the values a and d are the extremes . The values b and c are the means . • The product of the extremes and the product of the means are called the cross products . Theorems, Postulates, Corollaries, and Properties • Cross Products Property In a proportion, if =

a b

c d and b and d ≠ 0, then ad = bc . c d is equivalent to the following: =

a b

• Properties of Proportions The proportion

= ad bc b a d c a c = b d

A proportion can be expressed in different ways that all produce the same cross products because they all express the same relationship between the factors. Substituting numbers for the variables will show that the different forms all express the same relationship. Example 1 Writing Ratios

The slope of a line is expressed here as a ratio. The coordinates of the endpoints of the line are given. The slope of a line is defined as the ratio of the rise of the line (change in y ) to the run of the line (change in x ). The slope is the difference in y values ( y 2 − y 1 ) over the difference in x values ( x 2 − x 1 ). To express the slope of the given line, substitute the appropriate coordinate values into the formula for slope and simplify. The solution yields a slope of 1/4.

7.1.1 Ratio and Proportion (continued)

Example 2 Using Ratios

The ratio of the lengths of the sides of a quadrilateral is used to solve for the length of the longest side in this example. It is given that the sides are in a ratio of 2:3:4:9 and that the perimeter is 126 ft. To solve for the actual length of the longest side, set x to equal the common factor of the side lengths. The actual side lengths can be expressed as the common factor times the ratio number. Write an equation that sets the sum of the side lengths equal to the perimeter and solve for the common factor x . The solution yields x = 7 feet. This is not the answer to the example, which asked for the actual length of the longest side. The length of the longest side is 9 x = 9(7) = 63 feet. A proportion with an unknown is solved here for the value of the unknown. The given proportion is = w 3 60 40 . To solve for the unknown, w , take the cross product of the proportion and solve for w . The solution yields w = 2, meaning that 3 is to 2 as 60 is to 40. This is one of the most useful mathematical procedures to solve simple math questions in everyday life. A proportion with an unknown is solved for the value of the unknown in this example. The given proportion is + = + x x 3 5 20 3 . To solve for the unknown, x , take the cross product of the proportion and solve for x . There are two possible solutions for the unknown: x = 7 or x = − 13.

Example 3 Solving Proportions

7.1.1 Ratio and Proportion (continued) Example 4 Using Properties of Proportions

The ratio of two unknowns in an algebraic equation is determined in this example. To find the ratio of the two unknowns, manipulate the equation so that x and y are together in the form of a ratio. Simplifying yields the proportion = ,

x y

3 2

which includes the ratio x to y .

Example 5 Problem-Solving Application

A proportion with an unknown is solved for the value of the unknown in this application example. The given proportion is = h 3 1 15 . The proportion sets the ratio of the height to the width of a model chair equal to the ratio of the unknown height to the width of a full-size chair. To solve for the unknown height, h , take the cross product of the proportion and solve for h . The solution yields h = 45. The height of the full-size chair is 45 inches.

7.1.1 Ratio and Proportion - Worksheet

Example 1: Write a ratio expressing the slope of each line. 1. l 2. m

3. n

–4

–2

–4

Example 2: 4. The ratio of the side lengths of a quadrilateral is 2:4:5:7, and its perimeter is 36 m. What is the length of the shortest side?

5. The ratio of the angle measures in a triangle is 5:12:19. What is the measure of the largest angle?

Example 3: Solve each proportion. 6. = x 2 40 16

= 6 58 29 t

= y 7 21 27

= x x 18 6 2

y 3

−

10.

11.

−

Example 4: 12. Given that 2 a = 8 b , find the ratio of a to b in simplest form.

13. Given that 6 x = 27 y , find the ratio y : x in simplest form.

7.1.1 Ratio and Proportion - Worksheet (continued)

Example 5: 14. The Arkansas State Capitol Building

is a smaller version of the U.S. Capitol Building. The U.S. Capitol is 752 ft long and 288 ft tall. The Arkansas State Capitol is 564 ft long. What is the height of the Arkansas State Capitol?

7.1.1 Ratio and Proportion - Practice

1. The ratio of the side lengths of a quadrilateral is 2:7:8:12, and the perimeter is 522 cm. What is the measure of the shortest side?

2. Solve the proportion. = x x 5 20

3. Solve the proportion. + = − s s 5 9 16 5

4. Given that 12 x = 15 y , find the ratio of x to y in simplest form.

6. A scale model of a tower is 24 inches tall and 18 inches wide. If the height of the actual tower is 60 feet, what is its width?

5. A regulation-sized gymnasium is 80 feet wide and 100 feet long. If a scale model of the gymnasium is 12 inches long, how wide is the model?

7.1.2 Ratios in Similar Polygons Key Objectives • Identify similar polygons. • Apply properties of similar polygons to solve problems. Key Terms • Figures that are similar ( ∼ ) have the same shape but not necessarily the same size.

• A similarity ratio is the ratio of the lengths of the corresponding sides of two similar polygons. • Two polygons are similar polygons if and only if their corresponding angles are congruent and their corresponding sides are proportional. Example 1 Describing Similar Polygons

The pairs of congruent sides and angles in two similar triangles are identified in this example. The lengths of the sides are given. Two pairs of angles are given as congruent. According to the Third Angles Theorem, since ∠ B ≅ ∠ E and ∠ C ≅ ∠ F , then the third angles, ∠ A and ∠ D , are also congruent. To determine whether the sides are congruent, calculate the ratios of the lengths of each hypothesized congruent pair. When setting up the ratios, make sure to put all the lengths from one triangle in the same position, on the top or on the bottom. These ratios are all equal to 2, so all three pairs of sides in the triangles are congruent. In this example, two rectangles are compared to determine whether they are similar. The similarity ratio is calculated. It is given that both figures are rectangles. The lengths of two consecutive sides for each rectangle are given. Since both figures are rectangles, all the angles are right angles and therefore congruent. To determine whether the sides are congruent, calculate the two ratios of side lengths. Both ratios of side lengths are 3/2, so the rectangles are similar (since these are rectangles, the other two sides are congruent pairs with the given sides). The similarity ratio for the two rectangles is the common ratio of the lengths of the sides, 3/2. Rectangle XYZW ∼ rectangle ABCD .

Example 2 Identifying Similar Polygons

7.1.2 Ratios in Similar Polygons (continued)

In this example, two triangles are compared to determine whether they are similar. The similarity ratio is calculated. The lengths of all three sides for each rectangle are given. It is given that the triangles share one congruent angle. To determine whether the triangles are similar, calculate the ratios of lengths of each hypothesized congruent pair of sides. Two of the sides have lengths in a ratio of 4/3. The third sides have lengths in a ratio of 3/2. Because the ratios are not the same, the sides are not all proportional, and these are not similar triangles.

Example 3 Architecture Application

A scale drawing of a rectangular room is compared with the actual rectangular room in this application example. The unknown length of a side of the actual room is determined. The lengths of two sides of the drawing of the rectangular room are given. The length of one side of the actual rectangular room is given. To determine the unknown length of the side of the kitchen, set up the proportion using the ratios of the side lengths for the kitchen and the drawing. In this case, the proportion is: the length in the drawing is to the length of the kitchen ( x ) as the width in the drawing is to the width of the kitchen. Cross multiply the proportion and solve for the value of x , the length of the kitchen. The solution yields an approximate length of 7.7 feet for the length of the kitchen.

7.1.2 Ratios in Similar Polygons - Worksheet

Example 1: Identify the pairs of congruent angles and corresponding sides. 1. 2.

P 3 2 4

97°

4 4 97° 97°

97°

Example 2: Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 3. rectangles ABCD and EFGH A B 135 4. △ RMP and △ UWX M W

Example 3: 5. The town of Goodland, Kansas, claims that it has one of the worldʼs largest easels. It holds an enlargement of a van Gogh painting that is 24 ft wide. The original painting is 58 cm wide and 73 cm tall. If the reproduction is similar to the original, what is the height of the reproduction to the nearest foot?

7.1.2 Ratios in Similar Polygons – Practice

1. △ ABC is similar to △ MNL . Identify the corresponding congruent angles.

2. Determine whether the polygons are similar. If so, write the similarity ratio and the similarity statement.

3. Determine whether the polygons are similar. If so, write the similarity ratio and the similarity statement.

4. The length of a bedroom in a blueprint is 5 in. and the width is 7 in. The actual length of the bedroom is 12 ft. Find the actual width of the bedroom in feet.

5. The ratio of a model building’s dimensions to the building’s actual dimensions is 1/52. If the height of the model is 15 inches, what is the height of the actual building in feet ?

6. Indicate whether the following statement is sometimes, always, or never true. Two isosceles triangles are similar.

7.1.3 Triangle Similarity: AA, SSS, and SAS Key Objectives • Prove certain triangles are similar by using AA, SSS, and SAS. • Use triangle similarity to solve problems. Theorems, Postulates, Corollaries, and Properties • Angle-Angle Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. • Side-Side-Side Similarity Theorem If the three sides of one triangle are proportional to the three corre- sponding sides of another triangle, then the triangles are similar. • Side-Angle-Side Similarity Theorem If two sides of one triangle are proportional to two sides of anoth- er triangle and their included angles are congruent, then the triangles are similar. • Reflexive Property of Similarity △ ABC ∼ △ ABC • Symmetric Property of Similarity If △ ABC ∼ △ DEF , then △ DEF ∼ △ ABC . • Transitive Property of Similarity If △ ABC ∼ △ DEF and △ DEF ∼ △ XYZ , then △ ABC ∼ △ XYZ . Example 1 Using the AA Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

The reason why two triangles are similar is determined in this example. It is given that each triangle has a right angle. Angles B and D are congruent because they are both right angles, as given. According to the Vertical Angles Theorem, ∠ BCA ≅ ∠ DCE . Therefore, by Angle-Angle Similarity the two triangles are similar.

Example 2 Verifying Triangle Similarity

If three sides of one triangle are proportional to the three corresponding sides of another triangle, then the triangles are similar.

7.1.3 Triangle Similarity: AA, SSS, and SAS (continued)

Two triangles are determined to be similar in this example. The lengths of the sides of the triangles are given. Calculate the ratios of the pairs of corresponding sides of the two triangles. Each of the ratios of the sides is 3/2, so the triangles are similar, according to the Side-Side-Side Similarity Theorem. If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. Two triangles are determined to be similar in this example. The lengths of two corresponding sides of the triangles are given. The triangles share a common angle. By the Reflexive Property of Congruence, ∠ D ≅ ∠ D . Calculate the ratios of the pairs of corresponding sides of the two triangles. Each of the two ratios of the sides is 6. ∠ D is the angle included between the two proportional sides, so the triangles are similar, according to the Side-Angle- Side Similarity Theorem.

Example 3 Finding Lengths in Similar Triangles

The length of an unknown side in a triangle is found in this example using the similarity ratio for two triangles. It is given that PQ || ST , and the lengths PR , QR , and RT are given. According to the Alternate Interior Angles Theorem, ∠ Q ≅ ∠ S (because PQ || ST ). By the Vertical Angles Theorem, ∠ PRQ ≅ ∠ TRS . Therefore, △ PQR ∼ △ TSR by Angle-Angle Similarity. Now, find the unknown side length SR by applying the similarity ratio for the triangles. Set up a proportion using the known corresponding lengths RT and PR and the corresponding lengths SR and QR . Substitute the known lengths of sides, cross multiply, and solve the resulting equation for the length of SR . The solution yields SR = 36.

7.1.3 Triangle Similarity: AA, SSS, and SAS (continued) Example 4 Writing Proofs with Similar Triangles

Two triangles are proved to be similar in this example. Since Q is the midpoint of PR and T is the midpoint of PS as given, then PQ = QR and PT = TS . By the Segment Addition Postulate, PR = PQ + QR and PS = PT + TS . Substituting gives PR = PQ + PQ and PS = PT + PT . These equations are rearranged to find the ratios of two corresponding sides in the two triangles. Both these ratios are equal to 2. The angles between the proportional sides of the two triangles are congruent, because they are the same angle, P . Therefore, the triangles are similar by Side-Angle-Side Similarity. Two triangles are proved to be similar and the length of an unknown side is derived from the similarity ratio. It is given that LC || SE . The lengths of IS , LC , and SE are given. By the Corresponding Angles Theorem, ∠ ILC ≅ ∠ ISE (or ∠ ICL ≅ ∠ IES ). Because they are the same angle, ∠ I ≅ ∠ I . Therefore, by Angle- Angle Similarity, △ SEI ∼ △ LCI . Find the unknown side length, LI , by applying the similarity ratio for the triangles. Set up a proportion using the known lengths SI and SE , and the corresponding lengths LI and LC . Substitute the known lengths of sides, cross multiply, and solve the resulting equation for the length of LI . The solution yields LI = 4 cm.

Example 5 Problem-Solving Application

7.1.3 Triangle Similarity: AA, SSS, and SAS - Worksheet

Example 1: Explain why the triangles are similar and write a similarity statement. 1. A 2. P

47°

52° 81°

52°

Example 2: Verify that the triangles are similar. 3. △ DEF and △ JKL

4. △ MNP and △ MRQ

Example 3: Explain why the triangles are similar and then find each length. 5. AB A 6. WY U

8.75

7.1.3 Triangle Similarity: AA, SSS, and SAS - Worksheet (continued)

Example 4: 7. Given:

8. Given: SQ = 2 QP , TR = 2 RP Prove: △ PQR ~ △ PST

MN KL Prove: △ JMN ~ △ JKL J N M

9. The coordinates of A , B , and C are A (0, 0), B (2, 6), and C (8, − 2). What theorem or postulate justifies the statement △ ABC ~ △ AEF , if the coordinates of E and F are twice the coordinates of B and C ?

Example 5: 10. In order to measure the distance AB across the meteorite crater, a surveyor at S locates points A, B, C, and D as shown. What is AB to the nearest meter? nearest kilometer?

733 m

586 m

533 m

644 m

800 m

7.1.3 Triangle Similarity: AA, SSS, and SAS - Practice

1. Given that the two triangles are similar, which of the following is an incorrect similarity statement?

2. Determine whether the two triangles are similar. Explain.

○ △ GHI ~ △ QRP ○ △ HIG ~ △ RPQ

○ △ IHG ~ △ PQR ○ △ GIH ~ △ QPR

4. Write the similarity statement for △ ADE and △ BCE . Explain.

3. Determine whether the two triangles are similar.

6. Explain why the triangles are similar. Then find OP and MN .

5. Given that B is the midpoint of AC , and E is the midpoint of AD , prove △ ABE and △ ACD are similar.

7.1 Review Worksheet

7.1.1 Ratio and Proportion Write a ratio expressing the slope of each line. 1. l 2. m

3. n

4. The ratio of the side lengths of an isosceles triangle is 4:4:7, and its perimeter is 52.5 cm. What is the length of the base of the triangle?

5. The ratio of the angle measures in a parallelogram is 2:3:2:3. What is the measure of each angle?

Solve each proportion. 6. = y 6 8 9

x 14

z 12

50 35

3 8

y 5 16

+ 2 2 3 m

11. + x

+ 12 2 2 m

125

− 5

10.

12. Given that 5 y = 25 x, find the ratio of x to y in simplest form.

13. Given that 35 b = 21 c, find the ratio b:c in simplest form.

14. Madurodam is a park in the Netherlands that contains a complete Dutch city built entirely of miniature models. One of the models of a windmill is 1.2 m tall and 0.8 m wide. The width of the actual windmill is 20 m. What is its height?

7.1 Review Worksheet (continued)

7.1.2 Ratios in Similar Polygons Identify the pairs of congruent angles and corresponding sides. 15.

16.8

M 10

16.

Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 17. △ RSQ and △ UXZ R 18. rectangles ABCD and JKLM A M J B 18 24 32

37°

D C

53°

Q X

19. The ratio of the model carʼs dimensions to the actual carʼs dimensions is 1 56 . . The model has a length of 3 in. What is the length of the actual car?

7.1 Review Worksheet (continued)

7.1.3 Triangle Similarity: AA, SSS, and SAS Explain why the triangles are similar and write a similarity statement. 20. J 21.

74°

32°

Verify that the given triangles are similar. 22. △ KLM and △ KNL L

23. △ UVW and △ XYZ

1 2

Explain why the triangles are similar and then find each length. 24. AB B 25. PS

17.5

7.1 Review Worksheet (continued)

26. Given: CD = 3 AC, CE = 3 BC Prove: △ ABC ∼ △ DEC

= QR NR Prove: ∠ 1 ≅ ∠ 2 M P PR MR 1

27. Given:

28. The picture shows a person taking a pinhole photograph of himself. Light entering the opening reflects his image on the wall, forming similar triangles. What is the height of the image to the nearest tenth of an inch?

15 in.

5 ft 5 in.

4 ft 6 in.

7.2

Applying Similarity

7.2.1 Applying Properties of Similar Triangles Key Objectives

• Use properties of similar triangles to find segment lengths. • Apply proportionality and triangle angle bisector theorems. Theorems, Postulates, Corollaries, and Properties • Triangle Proportionality Theorem If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. • Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. • Two-Transversal Proportionality Corollary If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. • Triangle Angle Bisector Theorem An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. Example 1 Finding the Length of a Segment

If a line parallel to a side of a triangle intersects the other two sides, then it divides those two sides proportionally. Look carefully at the proportion of segment lengths. The length of an unknown line segment is determined using the Triangle Proportionality Theorem in this example. It is given that PT || QS . Set up a proportion according to the Triangle Proportionality Theorem and substitute the known segment lengths. Cross multiply and solve for the length of the unknown segment, TS . The solution yields TS = 18/7.

Example 2 Verifying Segments are Parallel

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

7.2.1 Applying Properties of Similar Triangles (continued)

Two lines are determined to be parallel using the Converse of the Triangle Proportionality Theorem in this example. The lengths of line segments on the

divided sides of the triangle are given. To determine if DB divides EC and AC

proportionally, find the ratios of the lengths of the line segments around points B and D . The ratio of AB to BC is 4. The ratio of ED to DC is 4. Since the ratios are proportional, AE || BD by the Converse of the Triangle Proportionality Theorem.

Example 3 Art Application

If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. Look carefully at the proportion of the line segment lengths formed by the intersection of the transversals. The length of a line segment is determined using the Two-Transversal Proportionality Corollary in this example. It is given that PT || QU || RV || SW . The lengths of four line segments formed by the intersection of the transversals is given. To find the unknown length, set up a proportion involving the line segment of unknown length and the line segments with given lengths. Substitute the known segment lengths, cross multiply, and solve for QS . The solution yields QS = 13.456 cm.

7.2.1 Applying Properties of Similar Triangles (continued) Example 4 Using the Triangle Angle Bisector Theorem An angle bisector of a triangle divides the

opposite side into two segments whose lengths are proportional to the lengths of the other two sides.

The lengths of two line segments are determined using the Triangle Angle Bisector Theorem. The lengths of the unknown segments are given as algebraic expressions of an unknown, x . The lengths of the other two sides of the triangle are given. Given BD bisects ∠ ADC , write a proportion that correctly relates the ratio of the lengths of the two line segments formed by the bisector to the ratio of the lengths of the other two sides. Substitute the given values into the proportion, cross multiply, and solve for the unknown, x . The solution yields x = 7. To obtain the answer to the example, the lengths of the line segments AB and BC , substitute 7 into the equations for their lengths. This gives AB = 8 and BC = 10.

7.2.1 Applying Properties of Similar Triangles - Worksheet

Example 1: Find the length of each segment. 1. DG E

2. RN

Example 2: Verify that the given segments are parallel. 3. AB and CD B A

4. TU and RS

1.5

67.5

Example 3: 5. The map shows the area around Herald Square in Manhattan, New York, and the approximate length of several streets. If the numbered streets are parallel, what is the length of Broadway between 34th St. and 35th St. to the nearest foot?

36th St.

275 ft

240 ft

35th St.

250 ft

34th St.

7.2.1 Applying Properties of Similar Triangles - Worksheet (continued)

Example 4: Find the length of each segment. 6. QR and RS P

7. CD and AD

y – 1

2 y – 4

x – 2

x + 1

7.2.1 Applying Properties of Similar Triangles - Practice

1. Find the length of AD .

2. Find the length of BA .

4. Find the lengths of QP and PR .

3. Find the length of HO .

5. Find the length of BP .

6. Verify that ST || QR by using the Converse of the Triangle Proportionality Theorem.

7.2.2 Using Proportional Relationships Key Objectives • Use ratios to make indirect measurements. • Use scale drawings to solve problems. Key Terms • Indirect measurement is any method that uses formulas, similar figures, and/or proportions to measure an object. • A scale drawing represents an object as smaller than or larger than its actual size. • A drawing’s scale is the ratio of any length in the drawing to the corresponding actual length. Theorems, Postulates, Corollaries, and Properties • Proportional Perimeters and Areas Theorem If the similarity ratio of two similar figures is a b ,, then the ratio of their perimeters is a b , and the ratio of their areas is a b , or . 2 2 2

⎞ ⎠ ⎟

⎛ ⎝ ⎜

a b

Example 1 Measurement Application

The height of an object is determined from measurements on the ground in this application example. It is given that the student measuring the height is 5 ft 2 inches tall. It is also given that the inflatable ape throws a shadow 10 ft 6 inches long, while the student throws a shadow 3 ft long. Begin by converting all measurements to a single unit, inches. Then, prove that the triangles are similar. AB and DE are parallel because the rays of the Sun that form them are parallel. Therefore, the angles they form with the ground are congruent: ∠ B ≅ ∠ E . Because they are both right angles, ∠ F ≅ ∠ C . Therefore, by Angle-Angle Similarity △ ABC ∼ △ DEF . Set up a proportion with the lengths of the known sides and the unknown length (height of the ape). Substitute the known values for length, cross multiply, and solve for the unknown value. The height of the ape is 217 inches, or 18 ft 1 in.

7.2.2 Using Proportional Relationships (continued) Example 2 Solving for a Dimension

An actual distance is determined from a map distance and a scale in this application example. The map distance is given as 7 cm, and the scale of the map is given as 2.5 cm:850 m. To find the actual distance, set up a proportion using the scale ratio and the ratio of the unknown distance to the known map distance. Cross multiply and solve for the unknown actual distance. The actual distance is 2380 m.

Example 3 Making a Scale Drawing

A scale drawing of the base of a house is made in this application example. The dimensions of the actual house are given. The scale for the drawing is given as 2 cm:5 m. To find the dimensions of the drawing, write two proportions using the scale ratio and the ratio of the drawing length and width to the length and width of the actual house. For each proportion, cross multiply and solve for the unknown length. The length for the drawing of the house base is 8 cm. The width for the drawing of the house base is 16.8 cm.

7.2.2 Using Proportional Relationships (continued) Example 4 Using Ratios to Find Perimeters and Areas

If the similarity ratio of two similar figures is a b , then the ratio of their perimeters is , and the ratio

a b

2 2

⎞ ⎠ ⎟

⎛ ⎝ ⎜

a b

, or

of their areas is

The perimeter is different by a simple multiplier, while the area is different by the square of the multiplier. The perimeter and area of a triangle are determined here using the Proportional Perimeters and Areas Theorem. The perimeter, area, and length of one side are given for a triangle that is similar to the example triangle. The length of the corresponding side of the unknown triangle is given. To determine the perimeter of the unknown triangle, set up a proportion with the ratio of the corresponding sides of the triangles and the ratio of the known perimeter to the unknown perimeter. Cross multiply and solve for the unknown perimeter. The unknown perimeter is found to be 45 ft. To determine the area of the unknown triangle, set up a proportion with the ratio of the squares of the corresponding sides of the triangles and the ratio of the known area to the unknown area. Cross multiply and solve for the unknown area. The unknown area is found to be 162 ft 2 .

7.2.2 Using Proportional Relationships - Worksheet

Example 1: 1. To find the height of a dinosaur in a museum, Amir placed a mirror on the ground 40 ft from its base. Then he stepped back 4 ft so that he could see the top of the dinosaur in the mirror. Amir’s eyes were approximately 5 ft 6 in. above the ground. What is the height of the dinosaur?

5 ft 6 in.

4 ft

40 ft

Example 2: The scale of this blueprint of an art gallery is 1 in:48 ft. Find the actual lengths of the following walls. 2. AB 3. CD

B C

4. EF

5. FG

Example 3: A rectangular classroom is 10 m long and 4.6 m wide. Make a scale drawing of the classroom using the following scales. 6. 1 cm:1 m 7. 1 cm:2 m 8. 1 cm:2.3 m

Example 4: Given: rectangle MNPQ ~ rectangle RSTU 9. Find the perimeter of rectangle RSTU .

10. Find the area of rectangle RSTU .

7.2.2 Using Proportional Relationships - Practice

1. Marcy is 5 ft 5 in. tall. To find the height of a building, she measured her shadow and the building’s shadow. What is the height h of the building?

2. A man is 5 feet 9 inches tall. To find the height of a tree, the shadow of the

man and the shadow of the tree were measured. The length of the man’s shadow was 3 feet 6 inches. The length of the tree’s shadow was 10 feet 6 inches. What is the height of the tree?

4. A blueprint for a lighthouse uses a scale of 1 in:5 ft. The lighthouse in the blue print is 6 in. tall. How tall is the actual lighthouse?

3. A man is 6 feet 2 inches tall. To find the height of a tree, the shadow of the man and the shadow of the tree were measured. The length of the man’s shadow was 2 feet 1 inch. The length of the tree’s shadow was 3 feet 7 inches. What is the height of the tree?

6. △ PTS ~ △ DRV. Find the perimeter and area of △ PTS to the nearest tenth.

5. A blueprint for a house uses a scale of 1.4 cm:6.5 ft. The actual house is 52 ft tall. How tall is the house in the blueprint?

7.2.3 Dilations and Similarity in the Coordinate Plane Key Objectives • Apply similarity properties in the coordinate plane. • Use coordinate proof to prove figures similar. Key Terms • A dilation is a transformation that changes the size of a figure but not its shape. • A scale factor describes how much the figure is enlarged or reduced. Example 1 Computer Graphics Application

A dilation of a rectangle in the coordinate plane is made in this example. The coordinates of the vertices of the rectangle and the scale factor are given. To find the image, or dilated figure, multiply the coordinates for the vertices of the preimage by the scale factor and draw the new rectangle. To multiply the coordinates, multiply each x and y value with the scale factor. Plot the points of the new figure and draw the rectangle. Because the scale factor is greater than 1, the new figure is larger than the old figure. For example, A (0, 6) is dilated to A '(0 ⋅ (4/3), 6 ⋅ (4/3)) = A '(0, 8).

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