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### Chapter 5: Ratios, Rates & Proportions Section 5

Using Similar Figures

Anticipatory Set

- Buffaloes, I need some help.
- Look at the tiles in my kitchen. They are huge. I have a brilliant idea, lets remodel! However, I will only remodel with your help. I still want tile in my kitchen, but I want smaller pieces of tile. You’re my construction team! Let’s do this!

California Standards

- Number Sense 1.3: Use proportions to solve problems. Use Cross-Multiplication as a method for solving such problems.

Key VocabularyLanguage of the Discipline

- PROPORTION: An equation stating that two RATIOS are EQUAL.
- Examples: 1/2 =2/4 a/b = c/d, where b and d CANNOT equal ZERO
- POLYGONS: A closed plane figure formed by three or more line segments that DO NOT cross
- SIMILAR POLYGONS: A geometric occurrence where two polygons have corresponding angles that possess the same measure AND the lengths of the corresponding sides form equivalent ratios.
- CROSS PRODUCTS PROPERTY: When given two ratios, this property states that the CROSS PRODUCTS will EQUAL each other. If the two ratios have EQUAL cross products, they form a PROPORTION.
- INDIRECT MEASUREMENT: Examining Similar Polygons by using proportions to determine missing measures.

What is a PROPORTION?

- PROPORTION:
- A PROPORTION is an EQUATION stating that 2 RATIOS are EQUAL.
- Another way to test for PROPORTIONALITY is to use the Cross Products Property.
- Here, 2 ratios are set equal, values are multiplied diagonally, if BOTH resulting products are EQUAL you have a PROPORTION.
- If not EQUAL, the ratios are NOT PROPORTIONAL.

CROSS PRODUCTS PROPERTY

- a Mathematic Property will come in handy because that give the RULE or GUIDELINE on how to attack a problem.
- The CROSS PRODUCTS PROPERTY states that if two ratios form a proportion, the CROSS PRODUCTS are EQUAL. If two ratios have EQUAL Cross Products, they form a PROPORTION.
- There are two ways to look at PROPROTIONS.
- ARITHMETIC: 5/7 = 25/35

(5)(35) = (7)(25)

175 = 175

- ALGEBRAIC: a/b = c/d b and d CANNOT equal ZERO (0).

ad = bc

Finding A Missing Measure: Example 1

- Two Triangles exist and are similar. Find the value of T.
- The small triangle has two sides with a measure of 22 and 24 inches.
- The large triangle has similar sides of T and 36 inches.
- Highlight the side with both numbers with a yellow highlighter
- Highlight the side with the variable with a green highlighter.
- Using Proportions, we have:
- 22/24 = T/36
- (22)(36) = (24)(T)
- 33 = T

37 inches

22 inches

24 inches

55.5 inches

T

36 inches

Finding A Missing Measure: Example 2

- +Two Parallelograms exist and are similar. Find the value of P.
- ++The small parallelogram has two pairs of sides with measures of 13 and 19 cm
- +The large parallelogram has similar sides of P and 57 cm.
- Highlight the side with both numbers with a yellow highlighter
- Highlight the side with the variable with a green highlighter.
- +Using Proportions, we have:
- +13/19 = P/57
- +(13)(57) = (19)(P)
- +39 = P

13 cm

19 cm

P

57 cm

Finding A Missing Measure: Example 3

34 inches

- Two Trapezoids exist and are similar. Find the value of T.
- The small trapezoid has two sides with a measure of 50, one side of 34 and one of 44 inches.
- The large trapezoid has similar sides where one is T inches, two are 80 inches and the other is 70.4.
- Highlight the side with both numbers with a yellow highlighter
- Highlight the side with the variable with a green highlighter.
- Using Proportions, we have:
- 34/50 = T/80
- (34)(80) = (50)(T)
- 54.4 = T

50 inches

44 inches

T

80 inches

70.4 inches

Quick Review

- PROPORTIONS
- A pair of ratios that equal one another.
- Proportions can be solved using multiple methods.
- SIMLIAR FIGURES
- Similar Figures assumes that if two polygons are similar, a proportion can be formed between the two and you can solve using Cross Products Property.
- Hint: Analyze your geometric shape carefully, make certain that it is similar and labeled correctly to set proportions.
- Using CROSS PRODUCTS PROPERTY to Solve
- Cross Products Property states that a pair of Ratios are a PROPORTION when their cross products equal the same value.
- Remember that you are taking the NUMERATOR from one Ratio and MUTLIPLYING it by the DENOMINATOR of the other.
- Use this property and ALGEBRA to solve the missing value.
- Once the missing cross product is determined, DOUBLE CHECK to make certain it works in the original proportion.

Check for Understanding

- Please determine the BEST answer for the following expression.
- Carry out ALL work and calculations in your NOTES for later reference
- Please write your answer on your white boards and wait for the teacher’s signal.
- On the count of 3, hold up your wipe boards.

C4U Question #1Checking for Understanding

- Question #1:

-The 2 Triangles are Similar.

-What Proportion can be used to find the Missing Measure?

Select the BEST answer:

A. 12/16 = Y/16

B. Y/16 = 60/48

C. 48/36 = Y/16

D. 36/12 = 16/Y

Y

12 cm

16 cm

60 cm

36 cm

48 cm

C4U Question #2Checking for Understanding

- Question #2:

-The 2 Triangles are Similar.

-What Proportion can be used to find the Missing Measure?

Select the BEST answer:

A. 25/E = E/60

B. 5/E = 60/25

C. E/25 = 5/60

D. E/5 = 60/25

5 cm

E

25 cm

60 cm

48 cm

C4U Question #3Checking for Understanding

- Question #3:

-The 2 Triangles are Similar.

-What is the value of the Missing Measure?

Select the BEST answer:

A. R = 18.4 cm

B. R = 20.0 cm

C. R = 22.6 cm

D. R = 19.7 cm

R

14 cm

11 cm

60 cm

42 cm

33 cm

C4U Question #4Checking for Understanding

- Question #4:

-The 2 Parallelograms are Similar.

-What is the value of the Missing Measure?

Select the BEST answer:

A. Y = 38.8

B. Y = 40.6

C. Y = 39.8

D. Y = 41.4

12 cm

23 cm

21.6 cm

Y

Guided Practice

- Students will work on their book work, focusing only on the problems #1-7 page 241
- Work carefully, show your problem solving process, and double check all calculations.
- Use scratch paper to carry out your work.
- Once you have completed the assigned problems, please raise your pencil.
- The teacher will then check your work and release you to complete the independent practice.

Independent Practice

- Once you have been signed off and released to complete Independent Practice, please complete the following assignment:
- Finish the 5-5 Work book pages # 8-13
- Homework:
- Work book page 248
- Read the directions carefully
- Due tomorrow morning

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