Geometry Unit 5: Similarity Geometry Unit 5: Similarity · 2021. 1. 4. · Geometry Unit 5: Similarity Ms. Talhami 15 7. The measures of two corresponding altitudes of two similar

Geometry Unit 5: Similarity

Ms. Talhami 1


Name_________________


Ms. Talhami 2

Helpful Vocabulary

Word Definition/Explanation Examples/Helpful Tips


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Ratios & Proportions


Ratio

Rate (Unit Rate)

Proportion

Example # 1: If one store has 360 items and another store has 100 of the same items, express the ratio of the items. Example # 2: John earns $350 a week. His take-home pay, however, is $295. What is the ratio of his gross pay to his take-home pay? Example # 3: Francine paid $16 for her 12-month subscription to Better Homes and Gardens magazine. Express as a rate.

Example # 4: !"= #

$%

Example # 5: In one day you earn $75 for 8 hours of work. If you work 37.5 hours for the week, what will your weekly pay be?


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Practice 1. Write each ratio as a fraction in lowest terms.

a. 2 to 4 b. &'()

c. 6 : 18 d. 21 : 15

e. &(&*

f. 3 to 12

g. 7 : 4 h. &*&(

i. 20 : 16 j. 15 to 36

2. Write each of the following rates as a unit rate.

a. $,-./(0./

b. &$'/10234.!'.05164.

c. &(*7184.!39:5.

d. ((')/4;218.&*-9#4.

3. Solve each proportion and give the answer in simplest form.

a. 6 : 8 = x : 12 b. (<= *

#

c. #%= &&

$ d. 4 : x = 6 : 9

e. $#= (

' f.

).!&.'= &(

#

g. 2 ½ : 3 ½ = x : 2 h. 1 : 2 = x : 9

i. 4 to 8 = 15 to x j. 18 : x = 3 : 11

4. Solve by using a proportion. Round answers to the nearest hundredth if necessary. a. You jog 3.6 miles in 30 minutes. At that rate, how long will it take you to jog 4.8 miles?

b. You earn $33 in 8 hours. At that rate, how much would you earn in 5 hours?


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c. An airplane flies 105 miles in ½ hour. How far can it fly in 1 ¼ hours at the same rate of speed?

d. What is the cost of six filters if eight filters cost $39.92?

e. If one gallon of paint covers 825 sq. ft., how much paint is needed to cover 2640 sq. ft.?

More Complex Proportions

Solve the following proportions for x:

1. 2.

3.

4. 5.

6.

x10

= 320

20x= 1024

412

= xx+ 8

515

= xx+ 8

x12 - x

= 1030

3x+ 33

= 7x -15


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Similarity Word Definition/Explanation Examples/Helpful Tips

Similar

Angle and Side Relationships of Similar Triangles

Methods to Prove Triangles Are Similar


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Practice Recognizing Similar Triangles Determine if the following pairs of triangles are similar. If yes, by which method? a.

b.

c.

d.

Practice Missing Parts of Similar Triangles If the following pairs of triangles are similar, find the missing part. a.

b.

c.


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Mixed Similar Triangles Practice


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Take It a Step Further 1. Jeff asks the teacher is ASA is also a similarity criterion. The teacher says yes but it isn’t needed. Why isn’t it needed?

2. Who is correct? Explain your reasoning.

Extra Practice Recognizing Similar Triangles Determine if the triangles are similar. If they are similar, state the criterion and complete the similarity statement with corresponding parts.


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Extra Practice Finding Missing Values of Similar Triangles Given that the following triangles are similar, set up a ratio/proportion and solve for the missing values.


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Find the Missing Values of the Similar Triangles 1.

x = ________ y = ________

2.

x = ________ y = ________

3.

x = ________ y = ________

Discovering Measurements of Similar Triangles

Determine if the following pairs of triangles are similar. If so, state by which criterion.

y

10.8

x

6

12

10

T

G

R

S

Q

y

10

x

124

8

W

T

V

X

Uα

α

o

ox

36

y

16

12

21


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Measurements of Similar Triangles Perimeters are the __________ ratio of the sides. Ex. Ratio of the sides = 2 : 9 Ratio of the perimeters = Areas are the __________ ratio of the sides. Ex. Ratio of the sides = 4 : 7 Ratio of the areas = Volumes are the __________ ratio of the sides. Ex. Ratio of the sides = 2 : 5 Ratio of the volumes =

Practice Problems

1. If ΔABC ~ ΔDEF and the perimeter of ΔABC is 9, find x, y, and z.

2. If ΔHIJ ~ ΔKJL and the perimeter of ΔKJL is 7.5, find x, y, and z.


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3. If ΔABC ~ ΔA`B`C`, find x and y

4. If ΔABC ~ ΔA`B`C`, find x and y

5. If ΔABC ~ ΔDEF and the perimeter of ΔDEF is 29, find x, y, and z.

6. If ΔEFG ~ ΔABC and the perimeter of ΔABC is 81, find x, y, and z.


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7. The measures of two corresponding altitudes of two similar triangles are 6m and 14m. If the perimeter of the first triangle is 21, what is the perimeter of the second triangle.

8. The lengths of the sides of a triangle are 8, 20, and 24. The length of the longest side of a similar triangle is 12. Find the perimeter of the smaller triangle.

9. The lengths of the sides of a triangle are 5, 6, and 7. If the perimeter of a similar triangle is 36 feet, find the shortest length of the other triangle.

10. The measures of corresponding medians in two similar triangles are in the ratio of 2 : 3. What is the ratio of the area of the smaller triangle to the area of the larger triangle?

Extra Practice

1) Mary bought similar kites for her little sister and for herself. Corresponding sides of the two similar kites have a ratio of 5:9. If the area of the smaller kite is 32inches, what is the area of the larger kite?

2) The lengths of the sides of a triangle are 7, 9, and 12. If the perimeter of a similar triangle is 36.4, find the length of the longest side.

3) If the ratio of the volumes of two similar figures is 8:125, what is the ratio of the lengths of the sides?


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Mid-Segment Theorem Discovery M is the midpoint of AB N is the midpoint of CB AB = 10 CN = 3

Mid-Segment Theorem


Triangle Mid-Segment

Examples

1. Given DE is the length of the mid-segment, find AB.

2. Given DE, DF, and FE are the lengths of the mid-segments, find the perimeter of triangle ABC.


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Practice # 1 – 10: D is the midpoint of AB and E is the midpoint of BC: 1. If DE = 8, find AC.

2. If DE = 6, find AC.

3. If AC = 20, find DE.

4. If AC = 17, find DE.

5. If BE = 4, find EC.

6. If AD = 7, find DB.

7. If DB = 5, find AB.

8. If BC = 9, find EC.


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9. If mÐDBE = 35, find mÐBAC.

10. If mÐBCA = 35, find mÐDEB.

# 11 – 12: M, R, and T are midpoints of AB, BC and CA, respectively in DABC.

11. For each side of DABC, name a segment parallel to each side.

12. If AB = 22, BC = 12 and AC = 20, find: a) perimeter of DABC______ b) perimeter of DMRT______

# 13 – 15: D, E and F are midpoints of RT, TS, and SR respectively in DRTS. 13. If DE = 3y – 2 and TS = 4y + 4, find : a) y_____b)DF______c)TS______

14. If FE = x + 3 and RT = 4x – 7, find: a) x_____b)FE______c)RT______

15. If EFDT is a rhombus, EF = 2x – 2 and FD = 4x – 9, find: a) x_____b)EF______c)FD______ d) ST______e)RT______


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Do Now 1. In the following triangle, M, N, and P are the midpoints of the sides. Name a segment parallel to the one given.

CD || ____

PE || ____

2. Find CD.

3. Find IK.

4. Solve for x.

Simplifying Radicals

Helpful Tip: List the perfect squares up to 10

1. √12

2. √80

3. √32

4. √180


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Mean Proportional Word Definition/Explanation Examples/Helpful Tips

Mean Proportional

Examples of Mean Proportional

1. Find the geometric mean of 8 and 18.

2. Find the mean proportional of 5 and 57.8.

3. 15 is the geometric mean of 25 and what other number?


5. 32 is the mean proportional of 16 and what other number?


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Practice Mean Proportionals 6. Find the geometric mean of 2 and 6.


8. √72 is the geometric mean of 18 and what other number?


10. √315 is the geometric mean of 16 and what other number? Intro to Triangle Mean Proportionals

1) How many triangles are in the following image? Name each one using the vertices. 2) If m<A=60°, solve for m<ACD, m<DCB, and m<B. 3) Are any the triangles in the image similar? If so, which ones and by which method?


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HLLS Discovery

Using the same image from the Do Now: If AD=2 and DB=6, how can we determine the length of AC?

Still using the same image from the Do Now: If CB=8 and DB=4, how can we determine the length of AD?


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HLLS Mean Proportional Theorem

HLLS Examples 1. If AB = 8, and AC = 4, find AD.

2. If AC = 10 and AD = 5, find AB.

3. If AC = 6 and AB = 9, find AD.

4. If DB = 4 and BC = 10, find AB.


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SAAS Discovery

If AD=2 and DB=8, how can we determine the length of CD?

SAAS Mean Proportional Theorem


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SAAS Examples 1. If AD = 3 and CD = 6, find DB.

2. If AD = 4 and DB = 9, find CD.

Mixed HLLS and SAAS Problems 1. If AD = 3 and DB = 27, find CD.

2. If AD = 2 and AB = 18, find AC.

3. If DB = 8 and AB = 18, find BC

4. If AD = 3 and DB = 9, find AC.




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7. If CD = 10 and AD = 4, find DB.

8. If CD = 5 and AD = 5, find DB.

9. If AD = 2 and DB = 6, find AC.

10. If AD = 3 and DB = 24, find AC

11. If BC = 10 and AB = 25, find DB.

12. If AC = 4 and DB is 4 more than the length of AD, find CD.

13. If AD = 12 and DB is three less than the CD, find CD.

14. If AB is 4 times as large as AD and AC is 3 more than AD, find AD.


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15. If AD = x + 5, DB = x and CD = 6, find x.

16. If AD = 7, and DB = x and BC = 12, find x.

17. If AD = x, DB = 12 and AC = 8, find x.

18. If CD = 6, AD = 3 and DB = 5x - 3, find x.

Extra HLLS and SAAS Practice

# 1 – 3: Find the missing values. (If not a whole number, leave it in simplest radical form) 1.

x = ________ y = _________ z = ________

2.

x = ________ y = _________ z = ________

3.

x = ________ y = _________ z = ________

94

zyx

86

zyx

142

zyx


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# 4 – 10: Find the missing values. (If not a whole number, then round to two decimal places.) 4.

x = ________ y = _________ z = ________

5.

x = ________ y = _________ z = _______ w = ________

6.

x = ________ y = _________ z = ________

8.

x = ________ y = _________ z = ________

9.

x = ________ y = _________ z = ________

10.

x = ________ y = _________ z = ________

15

9

z

y

xw

20

8z

yx

10

8

z

y

x

15 8

zy

x

13

5

z

y

x 125

z

y

x

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Geometry Unit 5: Similarity Geometry Unit 5: Similarity · 2021. 1. 4. · Geometry Unit 5: Similarity Ms. Talhami 15 7. The measures of two corresponding altitudes of two similar