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Chapter 5 : Ratios, Rates & Proportions Section 5

Chapter 5 : Ratios, Rates & Proportions Section 5. Using Similar Figures. California Standards. Number Sense 1.2: Interpret and use ratios in different contexts. This application deals with Geometry.

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Chapter 5 : Ratios, Rates & Proportions Section 5

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  1. Chapter 5: Ratios, Rates & Proportions Section 5 Using Similar Figures

  2. California Standards • Number Sense 1.2: Interpret and use ratios in different contexts. • This application deals with Geometry. • Number Sense 1.3: Use proportions to solve problems. Use Cross-Multiplication as a method for solving such problems.

  3. Key Vocabulary • 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.

  4. What is a PROPORTION? • PROPORTION: • A PROPORTION is an EQUATION stating that 2 RATIOS are EQUAL. • Some people think of EQUIVALENT Fractions as PROPORTIONAL. • 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.

  5. CROSS PRODUCTS PROPERTY • With RATIOS and PROPORTIONALITY, a Mathematic Property will come in handy. Remember that properties 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

  6. Finding A Missing Measure: Example 1 • Two Triangles exist and are similar. Find the value of T. • Examine each triangle carefully. Here, we can create Proportions using the different sides. • 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. • Using Proportions, we have: • 22/24 = T/36 • (22)(36) = (24)(T) • 33 = T • DOUBLE CHECK • 22/24 = 33/36 • (22)(36) = (24)(33) • 792 = 792 37 inches 22 inches 24 inches 55.5 inches T 36 inches

  7. Finding A Missing Measure: Example 2 • +Two Parallelograms exist and are similar. Find the value of P. • + Examine each parallelogram carefully. +Here, we can create Proportions using the different sides. • +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. • +Using Proportions, we have: • +13/19 = P/57 • +(13)(57) = (19)(P) • +39 = P • +DOUBLE CHECK • +13/19 = 39/57 • +(13)(57) = (19)(39) • +741 = 741 13 cm 19 cm P 57 cm

  8. Finding A Missing Measure: Example 3 34 inches • Two Trapezoids exist and are similar. Find the value of T. • Examine each trapezoids carefully. Here, we can create Proportions using the different sides. • 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. • Using Proportions, we have: • 34/50 = T/80 • (34)(80) = (50)(T) • 54.4 = T • DOUBLE CHECK • 34/50 = 54.4/80 • (34)(80) = (54.4)(50) • 2,720 = 2,720 50 inches 44 inches T 80 inches 70.4 inches

  9. 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.

  10. 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 wipe boards and wait for the teacher’s signal. • On the count of 3, hold up your wipe boards.

  11. C4U Question #1 • 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

  12. C4U Question #2 • 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

  13. C4U Question #3 • 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

  14. C4U Question #4 • 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

  15. Guided Practice • Students will work on a worksheet/book work, focusing only on the problems assigned by the teacher. • 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.

  16. Independent Practice • Once you have been signed off and released to complete Independent Practice, please complete the following assignment:

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