Mathworksheetsgo.com worksheet
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Mathworksheetsgo.com worksheet
I. Model Problems. II. Practice Problems III. Challenge questions VI. Answer Key Web Resources Mathworksheetsgo.com recommends www.meta-calculator.com, a free online graphing calculator (graphs implicit equations, does advanced statics like T-tests and much more) © www.MathWorksheetsGo.com All Rights Reserved Commercial Use Prohibited © www.MathWorksheetsGo.com All Rights Reserved Commercial Use Prohibited Terms of Use: By downloading this file you are agreeing to the Terms of Use Described at http://www.mathworksheetsgo.com/downloads/terms-of-use.php . I. Model Problems: Ratios in Similar Polygons Definition of Similar Polygons Two Polygons are similar if and only if the corresponding angles are congruent and the corresponding side lengths are proportional. The mathematical symbol used to state that two polygon are similar is . D A B C F E The Corresponding Angles are: A and D , B and E , and C and F . The Corresponding Sides are: AB and DE , BC and EF , and CA and FD . If we state that ABC DEF then the following is true: A D B E C F AB BC CA DE EF FD Corresponding angles are congruent. Corresponding Side lengths are proportional. ABC DEF Notice the Corresponding Angles line up in the same order in the similarity statement. The Similarity Ratio should be ratio between the corresponding sides lengths of the triangles. Overall, the triangles are the same shape, meaning all angles have the same measure, but one triangle is bigger than the other. Example 1 For the two triangle, write a similarity statement, then determine what we can conclude based on the figures. What is the similarity ratio between the two triangles? J 8 I K 3 5 6 U 4 10 S P Answer: The similarity statement is KUS PJI based on the angles that are congruent. Additionally we can make the following conclusions: K P, U J , S I based corresponding angles being congruent. KU US KS PJ JI PI based on corresponding sides being proportional. To find the similarity ratio, insert the side lengths. 3 4 5 1 , so the similarity ratio is 1:2. 6 8 10 2 Example 2 Find the measure all missing sides and angles using the properties of similar polygons. What is the similarity ratio? WZYX EFDB 8 Answer: Based on the X Y B D Similarity statement and the 102 12 9 Prosperities of parallelograms, E F We know the corresponding 78 W Z Angles are congruent. So we can conclude: X Z F 102 Y E D 78 To find the missing sides, we can use the properties of parallelograms and the similarity theorems. XW XY BE BD Corresponding Sides are proportional in similar polygons 12 8 9 x Place in the length of each side into the equation. 12 x 72 Cross Product Property x6 Multiplication property of Equality. We can now conclude the lengths of the following sides: BD EF 6 WZ 8 YZ 12 DF 9 12 3 The Similarity Ratio is after reducing the fraction. (This can also be or 9 2 written as 3:2) Example 3 Determine if the triangles are similar. If the triangles are similar, what is the similarity ratio? Answer: If the triangles are similar, then M T following would be true. 7 N 8 12 R Y 13 9.33 0.857=0.857=0.923 all values are not Equal, so the triangles are not similar. 6 P RP RM MP TY YE ET 6 8 12 Fill in the value of length. 7 9.33 13 E N A Example 4 If MCB HAT , we know AT 10 , MC 16 , and CB 14 . What is the length of HA ? Answer Based on the statement above and that the corresponding angles and sides must go in the correct order, meaning that M H , C A, and B T . Also we know that the following corresponding sides are proportional. MC CB MB HA AT HT The circled proportion will be used to help us solve the problem. 16 14 HA 10 Set up the proportion. 160 14( HA) Cross Product Property Divide both sides by 14. Multiplication property of equality. 11.43 HA Worksheet problems are on the next page II. Practice Problems For the figures below, write the similarity statement and determine the conclusions regarding corresponding sides and angles. E A 1. 2. G F A H E D M C B P T L N 3. 4. MATH RULZ Draw your own figure. L P E F Directions: Determine if the following polygons are similar. If they are similar, write the appropriate similarity statement. What is the similarity ratio? 30 Y Z B 5. 6. 14 G 12 H F 18 24 J 10 I H 7 24 I A C 52 40 5 K D 9 J X W E 7. Rectangles ABCD and WXYZ AD 30, AB 64,WZ 50,WX 96 8. JMR and KNP J and N are right angles R 67, P 23, MJ 24, MR 26, RJ 10 KN 15, NP 36, and KP 39 Directions : Determine whether the statement is true or false. 9. Two right triangles are similar. 10. Two squares are similar. 11. If two polygons are congruent, then the polygons are similar. 12. If two polygons are similar, then the two polygons are congruent. Find the value of x in the figures below. 13. 3x-8 6 15 2x-3 14. ABCD EFGH AB 4, BC 3, EF x 3, FG 2 x 4 III. Challenge Problems 15. A ballroom is 100 feet long by 75 feet wide. On a blueprint for the ballroom, the room is 7 inches long. What is the width of the ballroom on the blueprint if it similar to the actual ballroom 16. The dimensions of the United States is 13937 kilometers in width and 2450 kilometers in length. You teacher wants you to draw a scale diagram on a 8 ½ inch by 11 inch piece of paper. She suggested that you use a scale of 1 inch equals 1200 kilometers. Will the map fit on the paper? Explain why or why not. If it does not fit, what scale should you use? IV. Answers Quad DACB Quad FGHE; Corresponding Angles:D F ; A G; C H ; B E Corresponding Sides : AMP DA AC CB BD FG GH HE EF ELT ; 2. Corresponding Angles:A E; M L; P T Corresponding Sides : NIP AM MP AP EL LT ET NEF ; 3. Corresponding Angles:N N ; I E; P F Corresponding Sides : Quad MATH NI IP NP NE EF NF Quad RULZ ; 4. Corresponding Angles:M R; A U ; T L; H Z Corresponding Sides : MA AT TH MH RU UL LZ RZ 5. The polygons are similar. They have a similarity ratio of 2:1. FGHIJ BAEDH 6. The polygons are not similar. 7. The quadrilaterals are not similar. 8.The triangles are similar. The similarity ratio is 3:2 JMR NPK 9. False 13. x 1 4 10. True 11. True 12. False. 14. x 5 15. The width on the blueprint of the ballroom will be 5.25 inches. 16. The map would not fit on the paper. The length would be greater than 11 inches long when scaled down. Increasing the scale to 1 inch to 1300 kilometers would make the map fit on the 8 ½ by 11 inch piece of paper.