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Chapter 5 : Ratios, Rates & Proportions Section 5

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Chapter 5 : Ratios, Rates & Proportions Section 5. Using Similar Figures. Anticipatory Set . Buffaloes, I need some help.

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anticipatory set
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
California Standards
  • Number Sense 1.3: Use proportions to solve problems. Use Cross-Multiplication as a method for solving such problems.
key vocabulary language of the discipline
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
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
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
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
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
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
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
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 1 checking for understanding
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 2 checking for understanding
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 3 checking for understanding
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 4 checking for understanding
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
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
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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