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GEOMETRY

UNIT 3 WORKBOOK

SPRING 2017

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Geometry Section 7.1 Notes: Ratios and Proportions Date: Learning Targets:

1. Students will be able to write ratios. 2. Students will be able to write and solve proportions.

Vocab. and Topics Definitions, Examples and Pictures

Ratio

Examples of Ratios

Example 1: a) The number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth. b) The country with the longest school year is China, with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.)

Extended Ratios

Examples of Extended Ratios

Example 2: a) In ΔEFG, the ratio of the measures of the angles is 5:12:13. Find the measures of the angles.

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b) The ratios of the angles in ΔABC is 3:5:7. Find the measure of the angles. Example 3: a. The ratio of the measures of the sides of a triangle is 4:5:7, and its perimeter is 160

centimeters. Find the measures of each side of the triangle.

b. The ratio of the measures of the sides of a triangle is 4:7:11, and its perimeter is 3300

meters. What are the measures of the sides of the triangle?

c. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 280

feet. What are the measures of the sides of the triangle?

Proportion

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Cross Product Property

Converse of the Cross Products

Property

Examples of Solving

Proportions

Example 3: Solve the proportion.

a) 6 918.2 y

= b) 4 5 26

3 6x − −

=

c) 7 1 15.5

8 2n −

=

Example 4: a) Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Monique’s school, predict the total number of students with a dog or a cat.

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b) randomly surveyed 50 students and found that 20 had a part-time job. If there are 810 students in Brittany's school, predict the total number of students with a part-time job.

Summary 1. You jog 3.6 miles in 30 minutes. At that rate, how long will it take you to jog 4.8 miles? 2. You earn $33 in 8 hours. At that rate, how much would you earn in 5 hours? 3. An airplane flies 105 miles in ½ hour. How far can it fly in 1 ¼ hours at the same rate of speed? 4. What is the cost of six filters if eight filters cost $39.92? 5. If one gallon of paint covers 825 sq. ft., how much paint is needed to cover 2640 sq. ft.? 6. A map scale designates 1” = 50 miles. If the distance between two towns on the map is 2.75 inches, how many miles must you drive to go from the first town to the second?

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7. Bob is taking his son to look at colleges. The first college they plan to visit is 150 miles from their home. In the first hour they drive at a rate of 60 mph. If they want to reach their destination in 2 ½ hours, what speed must they average for the remainder of their trip? 8. Four employees can wash 20 service vehicles in 5 hours. How long would it take 5 employees to wash the same number of vehicles?

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Geometry Name: _____________________________________ Section 7.1 Worksheet 1. FOOTBALL A tight end scored 6 touchdowns in 14 games. Find the ratio of touchdowns per game.

2. EDUCATION In a schedule of 6 classes, Marta has 2 elective classes. What is the ratio of elective to non–

elective classes in Marta’s schedule?

3. BIOLOGY Out of 274 listed species of birds in the United States, 78 species made the endangered list.

Find the ratio of endangered species of birds to listed species in the United States.

4. BOARD GAMES Myra is playing a board game. After 12 turns, Myra has landed on a blue space 3 times.

If the game will last for 100 turns, predict how many times Myra will land on a blue space.

5. SCHOOL The ratio of male students to female students in the drama club at Campbell High School is 3:4.

If the number of male students in the club is 18, predict the number of female students?

Solve each proportion.

6. 25 = 𝑥

40 7. 7

10 = 21

𝑥 8. 20

5 = 4𝑥

6

9. 5𝑥4

= 358

10. 𝑥 + 13

= 72 11. 15

3 = 𝑥 − 3

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12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters. Find the

measures of each side of the triangle.

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13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters. What are the measures of the sides of the triangle?

14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet. What are the

measures of the sides of the triangle?

15. The ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches. Find the measures of each side of the triangle.

16. TRIANGLES The ratios of the measures of the angles in △DEF is 7:13:16.

Find the measure of the angles

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Geometry Section 7.2 Notes: Similar Polygons Date: Learning Targets:

1. Students will be able to use proportions to identify similar polygons. 2. Students will be able to solve problems using the properties of similar polygons.

Similar Polygons

Similarity Statements

Examples of Similarity Statements

Example 1: a) If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides. b) If ΔGHK ~ ΔPQR, determine which of the following similarity statements is not true. A) ∠HGK ≅ ∠QPR B) C) ∠K ≅ ∠R D) ∠GHK ≅ ∠QPR

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Scale Factor

Examples of Scale Factor

Example 2: a) Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu: b) Tan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu: Example 3: a) The two polygons are similar. Find the values of x and y.

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b) The two polygons are similar. Solve for a and b .

Perimeters of

Similar Polygons

Examples Perimeters Similar

Polygons

Example 4: If ABCDE ~ RSTUV, find the scale factor of ABCDE to RSTUV and the perimeter of each polygon. b) If LMNOP ~ VWXYZ, find the perimeter of each polygon.

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Summary Quadrilateral ABCD EFGH. Find the following: 1. Their scale factor 2. The values of x, y, and z 3. The perimeters of the two quadrilaterals 4. The ratio of the perimeters Quadrilateral IJKL MNOP. Find the following: 5. What is the scale factor of quadrilateral IJKL to quadrilateral MNOP? 6. Find m∠ I. 7. Find m∠ L. 8. Find NO. 9. Find IJ.

F E

H

G

21 y

z

30

8 x

D

A B

C 20

10

P O

N M

28

16 135º

I

K

J

L 35

5k 15

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Geometry Name: _____________________________________ Section 7.2 Worksheet Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning. 1. 2. Each pair of polygons is similar. Find the value of x. 3. 4. 5. PENTAGONS If ABCDE ∼ PQRST, find the scale

factor of ABCDE to PQRST and the perimeter of each polygon.

6. SWIMMING POOLS The Minnitte family and

the neighboring Gaudet family both have in-ground swimming pools. The Minnitte family pool, PQRS, measures 48 feet by 84 feet. The Gaudet family pool, WXYZ, measures 40 feet by 70 feet. Are the two pools similar? If so, write the similarity statement and scale factor.

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7. PANELS When closed, an entertainment center is made of four square panels. The three smaller panels are congruent squares.

What is the scale factor of the larger square to one of the smaller squares?

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Geometry Section 7.3 Notes: Similar Triangles Date: Learning Targets:

1. Students will be able to identify similar triangles using the AA similarity Postulate and the SSS and SAS Similarity Theorems. 2. Students will be able to use similar triangles to solve problems.

Angle-Angle Similarity

Example 1: a) Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. b) Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning.

Side – Side - Side

Similarity

Side – Angle - Side Similarity

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Examples of Similarity in

Triangles

Example 2: Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. a) b) Example 3:

a) If ΔRST and ΔXYZ are two triangles such that 23

RSXY

= , which of the following

would be sufficient to prove that the triangles are similar? A) B) C) ∠R ≅ ∠S D) b) Given ΔABC and ΔDEC, which of the following would be sufficient information to prove the triangles are similar?

A) 43

ACDC

= B) m∠A = 2m∠D C) AC BCDC EC

= D) 54

BCDC

=

Reflexive Property of Similarity

Symmetric Property of Similarity

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Transitive

Property of Similarity

Examples of Algebra with Similarities of

Triangles

Example 4: a) Given / /RS UT , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT. b) Given / /AB DE , AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC. Example 5: a) Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 p.m. The length of the shadow was 2 feet. Then he measured the length of Sears Tower’s shadow and it was 242 feet at the same time. What is the height of the Sears Tower?

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b) On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 foot 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?

Post-it Summary :

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Geometry Name: _____________________________________ Section 7.3 Worksheet Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning. 1. 2.

ALGEBRA Identify the similar triangles. Then find each measure. 3. LM, QP 4. NL, ML

5. PS, PR 6. EG, HG

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7. INDIRECT MEASUREMENT A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow. a. Write a proportion that can be used to determine the height of the lighthouse. b. What is the height of the lighthouse?

8. CHAIRS A local furniture store sells two versions of the same chair: one for adults, and one for children.

Find the value of x such that the chairs are similar. 9. BOATING The two sailboats shown are participating in a regatta. Find the value of x.

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Geometry Section 7.4 Notes: Parallel Lines and Proportional Parts Date: Learning Targets:

1. Students will be able to use proportional parts within triangles. 2. Students will be able to use proportional parts with parallel lines.

Triangle

Proportionally Theorem

Examples using Triangle

Proportionally Theorem

Example 1: a) In , / /RST RT VU∆ , SV = 3, VR = 8, and UT = 12. Find SU. b) In , / /ABC AC XY∆ , AX = 4, XB = 10.5, and CY = 6. Find BY.

Converse of

Triangle Proportionality

Theorem

Examples of Converse of

Triangle Proportionality

Theorem

Example 2:

a) 1 In DEF, DH = 18, HE = 36, and DG = .2

GF∆ Determine whether / /GH FE .

Explain.

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b) In ∆WXZ, XY = 15, YZ = 25, WA = 18, and AZ = 32. Determine whether / /WX AY .

Mid-segment of a Triangle

Examples of Mid-segment of a

Triangle

Example 3: a) In the figure, DE and EF are midsegments of ΔABC. Find AB, FE, and m∠AFE. b) In the figure, DE and are midsegments of ΔABC. Find BC, DE, and m∠AFD.

Proportional parts of Parallel Lines

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Example of Proportional Pats of Parallel Lines

Example 4: In the figure, Larch, Maple, and Nuthatch Streets are all parallel. The figure shows the distances in between city blocks. Find the value of x.

Congruent Parts of Parallel Lines

Examples of Congruent Parts of Parallel Lines

Example 5: Find the values of x and y.

Summary

1. x = _____ y = _____ 2. x = _____ y = _____

x° y°

40° 60°

6x + 11

x + 23 3y - 9

2y + 6

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3. x = _____ 4. x = _____ y = _____ 5. z = _____ 6. x = _____ 7. Find the perimeter of ∆ABC.

25 21

x 3x - 6

x 2x + 1

y

42°

65°

z° 52

28

x

26

12

23 A

B

C

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Geometry Name: _____________________________________ Section 7.4 Worksheet 1. If AD = 24, DB = 27, and EB = 18, find CE.

2. If QT = x + 6, SR = 12, PS = 27, and TR = x - 4,

find QT and TR.

Determine whether 𝑱𝑲���� ∥ 𝑵𝑴�����. Justify your answer. 3. JN = 18, JL = 30, KM = 21, and ML = 35

4. KM = 24, KL = 44, and NL = 56 JN

𝑱𝑯 ���� is a midsegment of △KLM. Find the value of x. 6. 7.

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7. Find x and y. 8. Find x and y.

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Geometry Section 7.5 Notes: Parts of Similar Triangles Date: Learning Targets:

1. Students will be able to recognize and use proportional relationships of corresponding angle bisectors, altitudes and medians of similar triangles. 2. Students will be able to use the triangle bisector theorem.

Special Segments of Similar

Triangles (7.8)

Special Segments of Similar

Triangles (7.9)

Special Segments of Similar

Triangles (7.10)

Examples of Special Segments

in Similar Triangles

Example 1: a) In the figure, ΔLJK ~ ΔSQR. Find the value of x. b) In the figure, ΔABC ~ ΔFGH. Find the value of x.

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Triangle Angle Bisector

Examples of

Triangle Angle Bisector

Example 3: a) Find the value of x. b) Find the value of x.

Summary Find x. 1. 2. 3. 4. Find the value of each variable. 1. 2.

28

3. 4.

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Geometry Name: _____________________________________ Section 7.5 Worksheet Find the value of x. 1. 2. 3. 4. 5. If △JKL ∼ △NPR, 𝐾𝑀����� is an altitude of △JKL, 𝑃𝑇���� is an altitude of △NPR, KL = 28, KM = 18, and PT =

15.75, find PR.

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6. If △STU ∼ △XYZ, 𝑈𝐴���� is an altitude of △STU, 𝑍𝐵���� is an altitude of △XYZ, UT = 8.5, UA = 6, and ZB = 11.4,

find ZY.

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Geometry Section 7.6 Notes: Similarity Transformations Date: Learning Targets:

1. Students will be able to identify similarity transformations. 2. Students will be able to verify similarity after a similarity transformation.

Dilation

Similarity Transformation

Center of Dilation

Scale Factor of a Dilation

Pre-image

Types of Dilations

– Enlargement

Types of Dilations

–Reduction

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Examples of Dilations

Example 1: a) Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. b) Determine whether the dilation from Figure A to Figure B is an enlargement or a reduction. Then find the scale factor of the dilation. Example 2: a) A photocopy of a receipt is 1.5 inches wide and 4 inches long. By what percent should the receipt be enlarged so that its image is 2 times the original? What will be the dimensions of the enlarged image?

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b) Mariano wants to enlarge a picture he took that is 4 inches by 7.5 inches. He wants it to fit perfectly into a frame that is 400% of the original size. What will be the dimensions of the enlarged photo? Example 3: a) Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: M(–6, –3), N(6, –3), O(–6, 6) image: D(–2, –1), F(2, –1), G(–2, 2)

b) Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. original: G(2, 1), H(4, 1), I(2, 0), J(4, 0) image: Q(4, 2), R(8, 2), S(4, 0), T(8, 0)

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Summary 1. Determine if the following scale factor would create an enlargement, reduction, or isometric figure. a. 3.5 b. 2/5 c. 0.6 d. 1 e. 4/3 Given the point and its image, determine the scale factor. 2. A(3,6) A’(4.5, 9) 3. G’(3,6) G(1.5,3) 4. B(2,5) B’(1,2.5) 5. The sides of one right triangle are 6, 8, and 10. The sides of another right triangle are 10, 24, and 26. Determine if the triangles are similar. If so, what is the ratio of corresponding sides?

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Geometry Name: _____________________________________ Section 7.6 Worksheet Determine whether the dilation from A to B is an enlargement or a reduction. Then find the scale factor of the dilation. 1. 2. 3. 4. 5. 6.

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Graph the original figure and its dilated image. Then verify that the dilation is a similarity transformation. 7. Q(1, 4), R(4, 4), S(4, –1); 8. A(–4, 2), B(0, 4), C(4, 2);

X(–4, 5), Y(2, 5), Z(2, –5) F(–2, 1), G(0, 2), H(2, 1)

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Geometry Section 7.7 Notes: Scale Drawing and Models Date: Learning Targets:

1. Students will be able to interpret scale models. 2. Students will be able to scale factors to solve problems.

Scale Model / Scale Drawing

Examples of Scale Models/Drawings

Example1: MAPS The scale on the map shown is 0.75 inches : 6 miles. Find the actual distance from Pineham to Menlo Fields. Use a ruler. The distance between Pineham and Menlo Fields is about 1 1

16 or 1.0625 inches.

Method 1: Write and solve a proportion.

Method 2: Write and solve an equation. Example Use the map above and a customary ruler to find the actual distance between each pair of cities. Measure to the nearest sixteenth of an inch. 1. Eastwich and Needham Beach 2. North Park and Menlo Fields

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3. North Park and Eastwich 4. Denville and Pineham 5. Pineham and Eastwich Example 2: a) The distance between Boston and Chicago on a map is 9 inches. If the scale of the map is 1 inch: 95 miles, what is the actual distance from Boston to Chicago? b) The distance between Cheyenne, WY, and Tulsa, OK, on a map is 8 inches. If the scale of the map is 1 inch : 90 miles, what is the actual distance from Cheyenne to Tulsa?

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Example 3: SCALE MODEL A doll house that is 15 inches tall is a scale model of a real house with a height of 20 feet. a. What is the scale of the model?

b. How many times as tall as the actual house is the model?

Example 4: The length of a model train is 18 inches. It is a scale model of a train that is

48 feet long. Find the scale factor. Example 5: An artist in Portland, Oregon, makes bronze sculptures of dogs. The ratio of the height of a sculpture to the actual height of the dog is 2:3. If the height of the sculpture is 14 inches, find the height of the dog. Example 6: The span of the Benjamin Franklin suspension bridge in Philadelphia, Pennsylvania, is 1750 feet. A model of the bridge has a span of 42 inches. What is the scale factor of the model to the span of the actual Benjamin Franklin Bridge?

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Example 7: a) A miniature replica of a fighter jet is 4 inches long. The actual length of the jet is 12.8 yards. What is the scale of the model? How many times as long as the actual is the model jet? b) A miniature replica of a fire engine is 9 inches long. The actual length of the fire engine is 13.5 yards. What is the scale of the replica? How many times as long as the model is the actual fire engine? Example 8: a) Gerrard is making a scale model of his classroom on an 11-by-17 inch sheet of paper. If the classroom is 20 feet by 32 feet, choose an appropriate scale for the drawing and determine the drawing’s dimensions. b) Alaina is an architect making a scale model of a house in a 15-by-26 inch display. If the house is 84 feet by 144 feet, what would be the dimensions of the model using a scale of 1 in. : 6 ft?

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Geometry Name: _____________________________________ Section 7.7 Worksheet MAPS Use the map of Central New Jersey shown and an inch ruler to find the actual distance between each pair of cities. Measure to the nearest sixteenth of an inch. 1. Highland Park and Metuchen 2. New Brunswick and Robinvale 3. Rutgers University Livingston

Campus and Rutgers University Cook–Douglass Campus

4. AIRPLANES William is building a

scale model of a Boeing 747-400 aircraft. The wingspan of the model is approximately 8 feet 10 1

16 inches. If the scale factor of the model is

approximately 1:24, what is the actual wingspan of a Boeing 747-400 aircraft? 5. ENGINEERING A civil engineer is making a scale model of a highway on ramp.

The length of the model is 4 inches. The actual length of the on ramp is 500 feet. a. What is the scale of the model?

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b. How many times as long as the actual on ramp is the model? c. How many times as long as the model is the actual on ramp? 6. MOVIES A movie director is creating a scale model of the Empire State Building to use in a scene.

The Empire State Building is 1250 feet tall. a. If the model is 75 inches tall, what is the scale of the model? b. How tall would the model be if the director uses a scale factor of 1:75? 7. MONA LISA A visitor to the Louvre Museum in Paris wants to sketch a drawing of the Mona Lisa, a

famous painting. The original painting is 77 centimeters by 53 centimeters. Choose an appropriate scale for the replica so that it will fit on a 8.5-by-11-inch sheet of paper.