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

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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 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 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 34 inches 50 inches 44 inches T 80 inches 70.4 inches

Quick Review PROPORTIONS SIMLIAR FIGURES 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 #1 Checking for Understanding -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 -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 -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 -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