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

Key Vocabulary  PROPORTION: An equation stating that two RATIOS are EQUAL.  Examples: 1/2 =2/4a/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.  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.

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/db 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.  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 = inches 37 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. + 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 = cm 19 cm P 57 cm

Finding A Missing Measure: Example 3  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  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, inches 34 inches 44 inches T 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 wipe boards and wait for the teacher’s signal.  On the count of 3, hold up your wipe boards.

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 16 cm 12 cm 48 cm 36 cm 60 cm

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 E 5 cm 48 cm 60 cm 25 cm

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 11 cm 14 cm 33 cm 42 cm 60 cm

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 = cm 12 cm Y 21.6 cm

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.

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