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

22 inches37 inches

24 inches

T

36 inches

55.5 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

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

34 inches

44 inchesT

70.4 inches

80 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/16B. Y/16 = 60/48C. 48/36 = Y/16D. 36/12 = 16/Y

Y

16 cm

12 cm

48 cm

36 cm

60 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 = 60/25C. E/25 = 5/60D. E/5 = 60/25

E

5 cm

48 cm

60 cm

25 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 cmB. R = 20.0 cmC. R = 22.6 cmD. R = 19.7 cm

R

11 cm

14 cm

33 cm

42 cm60 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 = 38.8B. Y = 40.6C. Y = 39.8D. Y = 41.4

23 cm

12 cm

Y

21.6 cm

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