Name: ___________________________________ Geometry Homework Calendar Unit 6: Similar Triangles

To be eligible for a test retake, you must complete 90% of the homework on time

Daily Homework Grade

Notes taken on:

Date HW is due:

Day Notes Topic Homework On Time Late

4 3 2 2

/ / 1

Ratios, Proportions &

Geometric Mean

Day 1 Homework

/ / 2

Using Proportions

Similar

Polygons

Day 2 Homework

Study for

Quiz

/ / Quiz

Days 1-2

/ / 3 Triangle

Similarity Day 3

Homework

/ / 4 Review of

Day 3 Notes

Day 4 Homework (Review of

Day 3)

/ / 5 Proportionality

Theorems & Dilations

Day 5 Homework

/

/ Unit 6

Review

Finish Unit 6 Review

Check

answer key online

Test will be given:

/

Alg Rev is Due

/

*Unit 6 Test

Algebra Review

(due in 2 class pds)

On Time HW Score:

____ / ____

Retake eligible?

Yes No

Total HW Score:

____ / ____

Warm Up Grade:

____ / ____

*Spiral #5 Due on Test Day

Unit 6 Day 1 Notes: Ratios, Proportions, & Geometric Mean Geom

If a and b are two numbers or quantities and 𝑏 ≠ 0, then the ratio of a to b can be written as 𝑎

𝑏 or 𝑎: 𝑏.

Why is it important to state that 𝑏 ≠ 0? Ratios are usually expressed in simplest form. Two ratios that have the same simplified form are called

_____________________________________________. To simplify a ratio, the units must be same.

1 foot = ______ inches

1 yard = ______ feet

1 yard = ______ inches

1 mile = ______ feet

1 mile = ______ yards

1 mile = ______ inches

1 min = ______ seconds

1 hour = ______ mins

1 hour = ______ secs

1 day = ______ hours

1 day = ______ minutes

1 day = ______ seconds

1 gallon = ______ quarts

1 quart = ______ pints

1 gallon = ______ pints

1 pint = ______ cups

1 quart = ______ cups

1 gallon = ______ cups

1 pound = ______ oz

1 cup = ______ oz

Simplify the ratios:

1) 64 m : 6 m 2) 5 𝑓𝑡

20 𝑖𝑛 3)

2 𝑝𝑖𝑛𝑡𝑠

16 𝑜𝑧

Ratio Word Problem: The perimeter of a rectangle is 56 inches. The ratio of the length to the width is 6:1. Find the length and the width.

Extended Ratio: A comparison of more than two quantities. Example: The measures of the angles in ∆𝐶𝐷𝐸 are in the extended ratio of 1:2:3. Find the measures of the angles.

Proportion: A comparison of two equal ratios. 𝑎

𝑏=

𝑐

𝑑

Cross Products Property: In a proportion, the product of the extremes equals the product of the means.

𝑎

𝑏=

𝑐

𝑑 Example: Solve the proportion.

a) 5

10=

𝑥

16 b)

1

𝑦+1=

2

3𝑦

Properties of Radicals Product Property: √𝑎𝑏 = √𝑎 ∙ √𝑏 when a and b are positive quantities.

Quotient Property: √𝑎

𝑏=

√𝑎

√𝑏 when a and b are positive quantities.

Simplify the radicals:

1. √50 2. √108𝑥3 3. √1

20

Geometric Mean: The geometric mean of two positive numbers a and b is the positive number x that satisfies 𝑎

𝑥=

𝑥

𝑏. So, 𝑥2 = 𝑎𝑏 and in the end, 𝑥 = √𝑎𝑏.

Describe in words how to find the geometric mean of 𝑎 and 𝑏: Example: Find the geometric mean of 24 and 48. Find the geometric mean of the two numbers given. 1) 12 and 27 2) 18 and 54 3) 16 and 18

Unit 6 Day 2 Notes: Use Proportions & Similar Polygons Geom to Solve Problems Properties of Proportions

Reciprocal Property: If two ratios are equal, then their reciprocals are also equal.

If 𝒂

𝒃=

𝒄

𝒅, then

If you interchange the means of a proportion, then you form another true proportion.

If 𝒂

𝒃=

𝒄

𝒅, then

In a proportion, if you add the value of each ratio’s denominator to its numerator, then you form another true proportion.

If 𝒂

𝒃=

𝒄

𝒅, then

Example: In the diagram, 𝑀𝑁

𝑅𝑆=

𝑁𝑃

𝑆𝑇. Write four true proportions.

Example: ∆𝐷𝐸𝐹~∆𝑀𝑁𝑃. Find the value of x.

Example: Find 𝐵𝐴 and 𝐵𝐷.

𝑅

𝑆

𝑇

10 𝑥

𝑀

𝑁

𝑃

8 4

𝐴

𝐵

𝐶

𝐷 𝐸

𝑥

3

18

12 6

𝐸

𝐹 𝐷

𝑥 9

12

𝑀

𝑁

𝑃

12

16

24

Example: Find 𝑆𝑇 and 𝑉𝑇

Example: Find 𝑥 and 𝑦.

Two polygons are similar polygons if: 1. All corresponding angles are congruent AND 2. All corresponding side lengths are proportional.

Proportional means the ratio of all corresponding side lengths are equivalent.

Example: 𝐴𝐵𝐶𝐷~𝐸𝐹𝐺𝐻 List of Corresponding angles List of ratios of corresponding sides (𝐴𝐵𝐶𝐷 to 𝐸𝐹𝐺𝐻)

Example: In the diagram, ∆𝑅𝑆𝑇~∆𝑋𝑌𝑍. List all pairs of congruent angles, check that the ratios of corresponding side lengths are equal and write the ratios of corresponding side length of ∆𝑅𝑆𝑇 to ∆𝑋𝑌𝑍 in a statement of proportionality.

𝑆

𝑇

𝑉

𝑅

𝑈

12 8

28

𝑥

𝒙𝟐

𝒚𝟐

𝐴

𝐵

𝐶

𝐷

𝐸 𝐹 𝐺 𝟑𝟓 𝟐𝟏

𝟏𝟖

𝟐𝟒

𝐴 𝐵

𝐶 𝐷

𝐸 𝐹

𝐺 𝐻

𝑅 𝑆

𝑇

25 30

20

𝑋 𝑌

𝑍

15 18

12

Example: Determine if the polygons are similar. If they are, write a similarity statement and find the ratio and the scale factor of 𝑍𝑌𝑋𝑊 to 𝐹𝐺𝐻𝐽.

Perimeters of Similar Polygons If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

If one polygon has side length “a” and another similar polygon has side length “b,” what is the ratio of the perimeters of the first polygon to the second?

Example: In the diagram, 𝐴𝐵𝐶𝐷𝐸~𝐹𝐺𝐻𝐽𝐾. Find the ratio, scale factor, the value of x, and the perimeter of 𝐴𝐵𝐶𝐷𝐸.

Example: In the diagram, ∆𝐽𝐾𝐿~∆𝐸𝐹𝐺. Find the length of 𝐾𝑀.

𝑊

𝑋

𝑌

𝑍

20

15

30

25

𝐹

𝐽

16

12

𝐺

𝐻

24

20

𝐴 𝐵

𝐶

𝐷 𝐸

𝑥

10 𝐹 𝐺

𝐻

𝐽 𝐾

15

15

18

12

9

𝐿 𝐽

𝐾

𝑀 48

𝐸

𝐹

𝐺 𝐻

35

40

Unit 6 Day 3 Notes- Triangle Similarity Geometry

Angle-Angle (AA) Similarity If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Side-Side-Side (SSS) Similarity If the corresponding side lengths of two triangles are proportional, then the triangles are similar. Side-Angle-Side (SAS) Similarity If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar. Example: Determine if the triangles are similar. If they are, write a similarity statement and justify why they are similar.

𝐴

𝐵

𝐶

𝐷

𝐸

35°

𝐿

𝑀

𝑁

3

4

6

𝑄

𝑅

𝑃

4

6

7

𝑋

𝑌

𝑍

𝑆

𝑇

𝑈

5

12.9

15

4.3

43°

48°

𝑆

𝑇

𝑈

𝑄

𝑅

𝐹

𝐺

𝐻

𝐼

𝐽

6

9

27

18

14

42

A

B

C

E

F D

A

B

C

E

F D

A

B

C

E

F D

Example: A flagpole casts a shadow that is 50 feet long. At the same time, a woman standing nearby who is five feet four inches tall casts a shadow that is 40 inches long. How tall is the flagpole to the nearest inch? Example: Find the value of x that makes ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹. Example: Is either ∆𝐷𝐸𝐹 or ∆𝐺𝐻𝐽 similar to ∆𝐴𝐵𝐶? How do you know?

4

𝐴

𝐵

𝐶

8

𝑥 − 1

𝐷

𝐸

𝐹

12 18

3(𝑥 + 1)

𝐶 𝐴

𝐵

8

16

12

𝐷

𝐸

𝐹 12

6 9

𝐺

𝐻

𝐽

8

16

10

Unit 6 Day 5 Notes: Propotionality Thms & Dilations Geometry

Review from Day 2 Notes:

Example: In the diagram, 𝑄𝑆 || 𝑈𝑇, 𝑅𝑆 = 4, 𝑆𝑇 = 6, AND 𝑄𝑈 = 9. What is the length of 𝑄𝑅?

Example: Find the length of 𝑌𝑍.

NEW!! If three parallel lines intersect two transversals, then they divide the transversals proportionally. Which lines are the transversals? Proportionality Statement: Example: Find 𝐴𝐵.

𝑇 𝑈

𝑅

𝑄 𝑆

𝑌

36

𝑉 𝑊

𝑍

𝑋 35 44

15

16 𝐴

18

𝐵

𝑈 𝑊

𝑋 𝑉

𝑌

𝑍

𝑟 𝑠 𝑡

𝑚

𝑛

If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. What angle is bisected? What is the “opposite side?” What are the “other two sides?” Statement of Proportionality: Example: Find 𝑅𝑆.

We previously reviewed Translations, Reflections, and Rotations in Unit 4- Triangle Congruence. A dilation is a transformation that ______________ or _______________ a figure to create a similar figure. The notation used is (𝑥, 𝑦) → (𝑘𝑥, 𝑘𝑦) where k is the scale factor of a dilation. If 0 < 𝑘 < 1 the dilation is a reduction. If 𝑘 > 1 the dilation is an enlargement. Example: Draw a dilation of the figure with 𝑘 = 2.

Example: Draw a dilation of the figure with a scale factor of 1

2.

15

𝑥

𝑆

𝑃

7

𝑄

13

𝑅

𝐴

𝐷

𝐶 𝐵

Example: Determine if the dilation from Figure A to Figure B is a reduction or an enlargement. Then find the scale factor of the dilation.

Example: The coordinates of ∆𝐴𝐵𝐶 are given as 𝐴(2,1), 𝐵(3,5), and 𝐶(6,3). ∆𝐴𝐵𝐶~∆𝑋𝑌𝑍. Coordinates given are 𝑋(6,3) and 𝑌(9,15). What are the coordinates of Z?

A

4

B

6

A

7

B 3.5

Name: ______________________ Date: _______ Geom Unit 6 Review

Simplify the ratio: 1) 4 𝑓𝑒𝑒𝑡∶24 𝑖𝑛𝑐ℎ𝑒𝑠 2)

55 𝑔𝑎𝑙

11 𝑔𝑎𝑙 3)

9 𝑞𝑡

6 𝑔𝑎𝑙

4) In the diagram, ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹.

Find 𝐸𝐹: _____

5) 𝐽𝐿

𝐿𝐻=

𝐽𝐾

𝐾𝐺

Find x: ____ Find JL: ____

6) Show all work:

A. What Is the ratio from 𝐷𝐸𝐹𝐺 to 𝑃𝑄𝑅𝑆?

B. Find the scale factor of 𝐷𝐸𝐹𝐺 to 𝑃𝑄𝑅𝑆.

C. Find the value of x.

D. Find the value of y.

E. Find the value of z.

F. Find the perimeter of EACH polygon.

7)

8)

9) Find the geometric mean of 9 and 27. Be sure to keep the radical in simplest form.

10) Find the angle measures for a triangle if the extended ratio of the angles is 1: 2: 3.

11) The perimeter of ∆𝐻𝐼𝐽 is 33. The extended ratio of the sides is 2:4:5. Find the lengths of the sides.

12) At the same time of day, a man who is 70 inches tall casts a 45-inch shadow and his son casts a 30-inch shadow. Use similar triangles to determine the height of the man’s son.

A. Draw a picture:

B. Why are the triangles similar?

C. What is the height of the man’s son?

13) Set up the proportion, and solve for x:

14) List the three ways to prove triangles similar:

15) Determine whether the triangles are similar (YES or NO). IF YES, provide the reason and write a similarity statement.

Similar? Yes/No

Reason for Similarity

Similarity Statement

16) Solve for x and simplify the radical: 3𝑥2 – 17 = 43

17) The “Grow Frog” package indicates that the frog will grow to 4 times its original size when placed in water for 6 days. The diagram below shows the original from of height 5cm and the frog after the growing process is completed. In relation to the diagram, find the value of x.

18) Draw a dilation of triangle ABC with the given vertices and the scale of k. Label the new triangle DEF. For the triangles to fit, you must CHANGE THE GRAPH TO COUNT BY 2!!

19) A polygon has the given vertices. Graph the polygon. All side lengths of each should be in simplest radical form…no decimals on this test. 𝐴(−2, 0), 𝐵(0,4), 𝐶(6,3), 𝐷(5, −3).

AB= ____

BC= ____

CD= ____

DA= ____

Which sides are perpendicular?

Why? This is the most important part!!!

20) Draw a venn diagram to represent each of the following situations:

A. All rainy days are cloudy.

B. Some students take band and chorus.

C. No freshmen take Driver’s Ed.

21) Let 𝑝 = 𝑡ℎ𝑒 𝑡𝑜𝑟𝑡𝑜𝑖𝑠𝑒 𝑖𝑠 𝑏𝑙𝑢𝑒 and 𝑞 = 𝑡ℎ𝑒 𝑓𝑟𝑜𝑔 𝑖𝑠 𝑎 𝑏𝑜𝑦. Write the following symbolic statements as a sentence or the following statements as symbolic notation. A. ~𝑝 ⋁ 𝑞

B. 𝑝 ⋀ ~𝑞

C. The tortoise is not blue and the frog is a not a boy.

MAKE SURE SPIRAL #5 IS COMPLETED FOR TEST DAY TOO!!

Name: ________________________________________ Geom Unit 6 Day 5 Homework 1. Determine which two of the three triangles are similar and provide the reason why. Find the scale factor

for the pair from smallest to largest.a.

b.

2. Use the diagram shown to complete the statements.

a. 𝑚∠𝐷𝐺𝐸 = ______

b. 𝑚∠𝐸𝐷𝐺 = ______

c. 𝐹𝐷 = ______

d. 𝐺𝐷 = ______

e. 𝐸𝐺 = ______

f. Name three pairs of triangles that are similar to each other:

3. Solve for x.

a.

b.

c.

d.

4. Pine Tree. In order to estimate the height h of a tall pine tree, a student places a mirror on the ground and stands where she can see the top of the tree, as shown in the diagram. The student is 6 feet tall and is standing 3 feet from the mirror which is placed 11 feet from the base of the tree.

a. What is the height, h, (in feet) of the pine tree?

b. Another student also wants to see the top of the tree. The other student is 5.5 feet tall. If the mirror is to remain 3 feet from the student’s feet, how far from the base of the tree should the mirror be placed?

5. Draw a dilation of the figure using the given scale factor, k.

6. Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then find the

scale factor from A to B and the values of the variables.a.

b.

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Worksheet by Kuta Software LLC

Geometry

Unit 6 Day 4 HW (Day 3 Review)

Name___________________________________

Date________________©J m2\0w1t8N UKquYtuac \SWoCfZt\wBaCreeO FLbLPCg.j J FAzlHlH rrlibgShKtjsG urHeNsXeXrnvDeXdf.

State if the triangles in each pair are similar. If so, state how you know they are similar andcomplete the similarity statement.

1)

54

84

S

T

117

182

B

CA

ABC ~ ______

A) similar; SAS similarity; ASTB) not similarC) similar; SSS similarity; ASTD) similar; SSS similarity; ATS

2)

C

B

F

G

E

EFG ~ ______

A) similar; AA similarity; BECB) similar; AA similarity; ECBC) similar; SAS similarity; BECD) not similar

3) 63

57

33

B C

D

26

G

F

BCD ~ ______

A) similar; SSS similarity; FGBB) not similarC) similar; SSS similarity; GBFD) similar; AA similarity; BFG

4) 140

110

99

L M

N

27

U

V

LMN ~ ______

A) similar; SAS similarity; LVUB) similar; AA similarity; LVUC) similar; SAS similarity; VULD) not similar

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Worksheet by Kuta Software LLC

State if the triangles in each pair are similar. If so, state how you know they are similar andcomplete the similarity statement.

5)

12

14

68 °E

F

G

48

52

68 °RQ

P

RQP ~ ______

6)

44 °50 °CD

E

44 °50 °FG

H

FGH ~ ______

7)

1310

11J

K

L

52

40

44

T

S R

TSR ~ ______

8)

50

35

L

M

60

42

Q

R

P

PQR ~ ______

Solve for x. The triangles in each pair are similar.

9) ABC ~ AQP

20

3x - 10

Q

P

25

40

B

CA

10) UVW ~ NML

2418

N

M

L

156

9x + 18

U

VW

Name: _________________________________________ Geometry Unit 6 Day 3 Homework 1. Use the diagram to complete the statement.

a. ∆𝐴𝐵𝐶~______

b. 𝐴𝐵

=𝐸𝐹=

𝐶𝐴

c. ∠𝐵 ≅_____

d. 12=

8

e. 𝑥 = _____

f. 𝑦 = _____

2. Which triangles are similar to ∆𝐸𝐹𝐺? How do you know? a.

b.

3. Determine whether the triangles can be proved similar. If they are similar, write a similarity statement and state the reason by which you know they are similar.

a.

b.

c.

4. Find the coordinates of point Z so that ∆𝑅𝑆𝑇~∆𝑅𝑋𝑍. Do you see the triangles in the diagram?? What do the ordered pairs below tell you about the triangles???? LABEL: R(0,0)

S(0, 4) T(-8,0) X(0,2) Z(?,?)

5. Flag Pole. In order to estimate the height h of a flag pole, a 5 foot tall male student stands so that the tip of his shadow coincides (matches up) with the tip of the flag pole’s shadow. This scenario results in two similar triangles as shown in the diagram.

a. Why are the two overlapping triangles similar?

b. Using the similar triangles, write a proportion that models the situation and solve it for h.

6. If possible, find the values of the variables. (It will only be possible if the two triangles in the diagram are similar!)

Name: _______________________________ Geom Unit 6 Day 2 Homework 1. Complete the proportion. Follow the pattern!

a. If 7

10=

𝑥

𝑦, then

10

7=

b. If 6

𝑥=

24

𝑦, then

6

24=

c. If 3

𝑥=

9

𝑦, then

3+𝑥

𝑥=

d. If 𝑥

𝑦=

5

11, then

𝑥+𝑦

𝑦=

2. Use the diagram and the given information to find the indicated length.

a.

b.

c.

d.

e.

f.

3. Determine whether the polygons are similar. If they are, write a similarity statement and find the scale factor.

a.

b.

4. In the diagram, WXYZ ~ MNOP.

a. Find the scale factor of WXYZ to MNOP

b. Find the values of x, y, and z.

c. Find the perimeter of WXYZ.

d. Find the perimeter of MNOP.

e. Find the ratio of the perimeter of MNOP to the perimeter of WXYZ. 5. The two triangles are similar. Find the values of the variables.

Name: __________________________________ Geom Unit 6 Day 1 Homework 1. Simplify the ratio:

a. $12 : $16

b. 32 𝑖𝑛2

8 𝑖𝑛2

c. 6 𝑐𝑚

14 𝑐𝑚

d. 10 𝑖𝑛

2 𝑓𝑡

e. 3 gallons : 10 quarts

f. 28 oz : 2 lb

2. The perimeter and the ratio of the length to the width of a rectangle are given. Find the length and width of the rectangle.

a. Perimeter = 50 in 𝑙: 𝑤 = 3: 2

b. Perimeter = 480 ft 𝑙: 𝑤 = 5: 1

c. Perimeter = 36 cm 𝑙: 𝑤 = 8: 1

3. The area of a rectangle is 525 cm2. The ratio of the length to the width is 7 : 3. Find the length and the width.

4. The measures of the angles of a triangle are in the extended ratio given. Find the measures of the angles of the triangle.

a. 1 : 1 : 1

b. 1 : 1 : 2

c. 2 : 3 : 4

5. Find the geometric mean of the two given numbers. a. 2 and 8

b. 3 and 9

c. 7 and 14

d. 8 and 16

e. 10 and 12 f. 9 and 13

6. The ratio of the two side lengths for the triangle is given. Solve for the variable. HINT: Set up a proportion and solve it!

a.

b.