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CSUN MATH 310 - ANSWERS & COMMENTS

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Math 310 Ì SIM ILARITY Ì ANSWERS & COMMENTS F7 1 2 31.123 123 1&2 123 1&2 1&2 123 none 2&3 C E D I 1212. “SSS” 110 121 77 77 121 110 70 G 100 H A B F110 ALL sides congruent(121-110-77) Sides in same proportions as in ABC (11:10:7)So ªs ABC & DEF are congruent So ªGHI is similar to ªABCIF the three SIDES of ÎABC are CONGRUENT to the respective sides of ÎFED then ÎABC ÎFED .IF the lengths of the three sides of ÎABC and the corresponding sides of ÎGHI 121 110 77))) = ))) = )) = 11: 10all form the SAM E RATIO, then the tw o triangles are SIM ILAR. 110 100 70 C F3. “SAS” I100 100 8040E 40E 40E A 110 B D 110 E G 88 HIf tw o sides and the included angle of ÎABC are congruent to the corresponding sides and included angle of ÎDEF then the tw o triangles are congruent.If tw o sides of ÎABC and the corresponding sides of ÎGHI form the same ratio, and the 100 110 ))) = )))included angles are congruent, then the tw o triangles are similar. 80 88C F4. “ASA” “ AA” I “ Side information” NOT needed!40E 60E40E 60E 40E 60E G H AB D E110 1105. AAA ÿ similar triangles Recall: SSS ÿ congruent triangles butAAAA ÿ' similar quadrilaterals example here: SSSS ÿ' congruent quadrilaterals.AD E H I L ADBC has all angles 90E all IJKL has sides congruent to the sides but is not similar to Angles of EFGH, but rhombus IJKL Square EFGH, show n º 90E is NOT congruent to square EFGH.BC F G JKWhat is theSCALINGFACTOR? 10(/ )8Math 310 Ì SIM ILARITY cont’d Ì 6. Triangle ABC has congruent sides AB & AC. The midpoint of BC is "E". Is AE.an altitude of ªABC?What triangles are congruent, and how do w e know ? A AB= A C, BE= CE are given. AE= AE (shared side) ...soªABE  ªACE by the “ SSS” congruence theorem.What do w e know about pABE and pACE?Since ªABE  ªACE, pABE and pACE must be congruent. B C What do w e know about pAEB and pAEC? E Since ªABE  ªACE, pABE and pACE must be congruentSince they are supplementary, each must be 90E. 7. Given that: A B is t he m idpoint of segm ent A D. C is t he midpoint of A F. E is the midpoint of segment DF. What segments are parallel?B C BC 2 DF BE 2 AF CE 2 AD What triangles are congruent, and w hy? ACB, BED, CFE and EBC. (Discussed in class.) What triangles are similar, but not congruent? DE F The four triangles named above are similar to AFD.D 8. Points B & C “ cut” segment AD into thirds. C H Similarly F& I and H& J cut segments A K and DK into thirds. B G J W hat can be said about segment s BJ and AK? They are PARA LLEL ! W hat can be said about t riangles BJD and A KD? They are SIM ILA R.A F I K If AB is 5u, then, since B & C cut A D in thirds, BD must be 2@5u, i.e. 10u, and AD must be 15u.If BJ is 12 u, then AK must be 18u .15 9. a. A b. E D c.* 14 B7 C 13 ED 59 15 21 D 5 6 A C B CA B BC | | ED AB | | DE pA and pD are both right angles. EAll segments that appear straight are straight (including ACD and BCE in 2 and 3 sketches).nd rdWhat triangles are similar and w hy? Can you find the missing segment lengths?ªAED ~ ªABC AD = 8.4 ªDEC ~ ªABC ªDEC ~ ªABC AC= 12 BC = 180/7 AB = 15(3/2) = 45/2 DE = 9(5/12) CE = 9(12/13)SCALE FACTOR is 3/2 or 1.5There is not enough information givenTo find lengths of BC and EC IN-CLASS Word Problem Example (w ith the real measurements): A redw ood tree casts a shadow 70' long at the same time a 10' tall street sign casts an 8' shadow .How tall must the tree be?The sunlight streaming dow n, interrupted by the vertical tree and sign,forms the shadow s on the horizontal ground. These right triangles have congruent base angles, since the sunlight hits the ground at the same angle in each triangle! By the “ AA” theorem of ª similarity, the tw o triangles must be similar.10' 70' 8' H10 10' (70))) = )) û H = )))))))) = 87.5' 70' 8 8geo_simf7Math 310 Ì SIM ILARITY some problems Ì 6" 10. If held 10" from the w all, a spray gun paints a disc 3" in diameter. How far from the w all should the gun be held to spray a toy that is 6" w ide? 3"TOUGHER QUESTION: If at 10" we spray for 1 second, at the new distance, how long should we spray to get the same thickness of paint? If the w idth of the spray is 3" w ide at 10" distance, ??then it should be 6" wide w hen the spray gun is 20" aw ay. 10"When the w idth is doubled, the area is quadrupled, so the paint ends up thinner,by a factor of 4. So the spray must be continued four times as long— 4seconds. 11. The sun is low in the sky, and casts a sixteen-foot shadow for a five-foot tall person. At the same time, theshadow of the redw ood tree is 352 feet long. What is the height of the redw ood tree?See Example 1 H 5'))) = ))) for reasoning. 352' 16' 5 352' H = ))) H? 16 = 5@22' = 110' 5' 16' 352' 12. Tom, Billy and Ann live in a straight line. Tom lives four blocks nort h and tw elve blocks w est of Billy. Billy livessix blocks north of Ann. Explain exactly w here Ann lives relative to Billy. T12Once you view the information in an appropriate w ay (as in a map view ), 4The relationship becomes clear.... w ith T, B and A in a straight line, there must be similar triangles connecting B( ? )them on the map. 6The only thing w e don’t know is how many blocks w est of Ann Billy lives.( ? ) 12 blocks ( Solve t his f or “ ?” )))) = )))))))) 6 bl. 4 bl.Ann is 18 blocks east and 6 blocks south of Bily Ann 13. * A 6' tall man standing 3' from a lamppost casts a shadow 4.5' long. How long w ould his shadow be if he stood 6' aw ay from the lamp?First set up the proportion for the initial situation given.Find the height of the lamppost. L (3+ 4.5) ‘ 7.5))) = )))))))) = ))) 6 ‘ 4.5' 4.5 L 10' L = (5/3) 6' = 10' 6' 6'Then redraw the sketch and set up a new proportion toansw er the second question, w hen he is 6' from the lamppost.3' 4.5' 6' x …


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