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Ch 7 Similarity C. Merrill Reitz High School 2013 - 2014 Lesson 7-1 Using Proportions Ratio Ratio: A ratio is a comparison of two numbers such as a : b. When writing a ratio, always express it in simplest form. Example: What is the ratio of AB to CB ? AB 10 A CB 6 Now try to reduce the fraction. 10 5 6 3 ratio of AB to CB 5:3. 8 10 D 4.8 C 3.6 6 B Example………. A baseball player goes to bat 348 times and gets 107 hits. What is the players batting average? Solution: Set up a ratio that compares the number of hits to the number of times he goes to bat. Ratio: 107 348 Convert this fraction to a decimal rounded to three decimal places. Decimal: 107 0.307 348 The baseball player’s batting average is 0.307 which means he is getting approxiamately one hit every three times at bat. Proportion Proportion: An equation that states that two ratios are equal. Terms a c b d First Term Second Term Third Term Fourth Term To solve a proportion, cross multiply the proportion: a d c b Extremes: a and d Means : b and c Proportions- examples…. Example 1: Find the value of x. x 2 ft 356 yards 84 yards Example 2: Solve the proportion. 8x = 30 x 5 6 8 8•x = 6•5 x = 3.75 8x = 30 8 8 6 7 Lesson 7-2 Similar Polygons 8 Similar Polygons Definition: Two polygons are similar if: 1. Corresponding angles are congruent. 2. Corresponding sides are in proportion. Two polygons are similar if they have the same shape not necessarily have the same size. Scale Factor: The scale factor is the ratio between a pair of corresponding sides. Naming Similar Polygons When naming similar polygons, the vertices (angles, sides) must be named in the corresponding order. If ABCD PQRS A P ; B Q ; C R ; D S P AB BC CD AD A B PQ QR RS PS D Q C S R Example- A 15 x H The two polygons are similar. 20 Solve for x, y and z. B E D y 30 10 C 5 F z G AD DC BC AB Step1: Write the proportion of the sides. EH HG FG EF Step 2: Replace the proportion with values.15 y 30 20 x 5 z 10 Step 3: Find the scale factor between the two polygons. Note: The scale factor has the larger quadrilateral in the numerator and the smaller quadrilateral in the denominator. Step 4: Write separate proportions for each missing side and solve. 15 2 y 2 30 2 x 7.5 y 10 z 15 x 1 5 1 z 1 If ABC ~ ZYX, find the scale factor from Example: ABC to ZYX. Scale factor is same as the ratio of the sides. Always put the first polygon mentioned in the numerator. C 10 B 5 AB 18 2 Y X ZY 9 1 14 18 7 9 The scale factor from Z ABC to ZYX is 2/1. A What is the scale factor from ZYX to ABC? ½ Lesson 7-3 Proving Triangles Similar (AA, SSS, SAS) AA Similarity (Angle-Angle) If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar. E B A C Given: Conclusion: D A D F and B E ABC ~ DEF SSS Similarity (Side-Side-Side) If the measures of the corresponding sides of 2 triangles are proportional, then the triangles are similar. E B 5 A Given: 10 8 11 C D AB BC AC DE EF DF Conclusion: ABC ~ DEF 16 22 F 8 5 11 16 22 10 SAS Similarity (Side-Angle-Side) If the measures of 2 sides of a triangle are proportional to the measures of 2 corresponding sides of another triangle and the angles between them are congruent, then the triangles are similar. E B 5 A 10 11 C D AB AC Given: A D and DE DF Conclusion: ABC ~ DEF 22 F Proving Triangles Similar Similarity is reflexive, symmetric, and transitive. Steps for proving triangles similar: 1. Mark the Given. 2. Mark … Shared Angles or Vertical Angles 3. Choose a Method. (AA, SSS , SAS) Think about what you need for the chosen method and be sure to include those parts in the proof. Problem #1 Given: DE ll FG PROVE: Triangle DEC ~ Triangle FGC Step 1: Mark the given … and what it implies Step 2: Mark the vertical angles Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Step 5: Is there more? Statements Reasons 1. DE is parallel to FG G Given AA D C E F Problem #2 GIVEN: IJ = 3LN, JK = 3NP, IK = 3LP PROVE: Triangle IJK ~ Triangle LNP Step 1: Mark the given … and what it implies Step 2: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Statements Reasons Step 5: Is there more? 1. IJ = 3LN ; JK = 3NP ; IK = 3LP Given SSS J K N I L P Problem #3 GIVEN: G is midpoint of ED, H is midpoint of EF PROVE: Triangle EGH ~ Triangle EDF Step 1: Mark the given … and what it implies Step 2: Mark the reflexive angles SAS Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons E Step 5: Is there more? G H D F 20 Statements Reasons 1. G is the Midpoint of ED Given H is the Midpoint of EF 2. EG = DG and EH = HF Def. of Midpoint 3. ED = EG + GD and EF = EH + HF Segment Addition Post. 4. ED = 2 EG and EF = 2 EH Substitution ED EF Division Property 5. EG =2 and EH =2 ED EF 6. = EG EH Substitution 7. GEHDEF Reflexive Property 8. EGH~ EDF SAS Postulate 21 Lesson 7-4 Proportional Parts Similar Polygons Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. AB AC BC DE DF EF F C A B D E Side Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides in two distinct points, then it separates these sides into C segments of proportional length. If BD AE , then CB CD BA DE B 1 Converse: If a line intersects two sides of a triangle and A 4 separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. CB CD If BA DE 2 D 3 E , then BD AE Examples……… Example 1: If BE = 6, EA = 4, and BD = 9, find DC. 6 9 9 4 x D 6x = 36 x x=6 C Example 2: Solve for x. 2x + 3 E 5 A B 6 E 4 2x 3 4x 3 5 9 5(4 x 3) 9(2 x 3) B 4x + 3 20 x 15 18 x 27 D 9 2 x 12 C x6 A Midsegment Theorem A segment that joins the midpoints of two sides of a triangle is parallel to the third side of the triangle, and its length is one-half the length of the third side. R If L is the midpo int of RS and M is the midpo int of RT then 1 S LM ST and ML ST . 2 L M T Extension of Side Splitter If three or more parallel lines have two transversals, they cut off the transversals proportionally. If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every E transversal. D AB DE AC BC AC DF , , , etc. BC EF DF EF BC EF A B C F Forgotten Theorem An angle bisector in a triangle separates the opposite side into segments that have the same ratio as the other two sides. AD AC If CD is the bi sec tor of ACB, then DB BC C A D B If two triangles are similar: (1) then the perimeters are proportional to the measures of the corresponding sides. (2) then the measures of the corresponding altitudes are proportional to the measure of the corresponding sides.. (3) then the measures of the corresponding angle bisectors of the triangles are proportional to the measures of the corresponding sides.. AB BC AC Perimeter of DE EF DF Perimeter of D AG (altitude of DI (altitude of A B G H C E I ABC ~ DEF J F ABC DEF ABC ) DEF ) AH (angle bi sec tor of ABC ) DJ (angle bi sec tor of DEF ) Example: Given: ΔABC ~ ΔDEF, AB = 15, AC = 20, BC = 25, and DF = 4. Find the perimeter of ΔDEF. The perimeter of ΔABC is 15 + 20 + 25 = 60. Side DF corresponds to side AC, so we can set up a proportion as: AC Perimeter of DF Perimeter of 20 60 4 x 20 x 240 x 12 ABC DEF E B 25 15 A D C 20 F 4