Math 310 Ì SIM ILARITY Ì ANSWERS & COMMENTS S7 1.123 123 1&2 123 1&2 1&2 123 none 2&3 C E F I 1212. “SSS” 110 121 77 77 121 110 70 G 100 H A B D110 sides congruent Sides in same proportions as in ABC (11:10 :7)So ªs are congruent So ªGHI is similar to ªABCIF the three sides of ÎABC are congruent to t he respective sides of ÎFED then the tw o triangles are congruent.IF the lengths of the three sides of ÎABC and the corresponding sides of ÎGHI 121 110 77))) = ))) = )) = 11:10all form t he same RATIO , then the t w o triangles are similar. 110 100 70 C F3. “SAS” I100 100 8040° 40° 40° 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 two t riangles are congruent.If tw o sides of ÎABC and the corresponding sides of ÎGHI form t he sam e rat io, and the 100 110))) = )))included angles are congruent, then the two triangles are similar. 80 88C F4. “ASA” “AA” I “Side information” NOT needed!40° 60°40° 60° 40° 60° G H A B D E110 1105. AAA ÿ similar triangles Recall: SSS ÿ congruent t riangles butAAAA ÿ' similar quadrilat erals example here: SSSS ÿ' congruent quadrilat erals.A D E H I L ADBC has all angles 9 0° IJKL has sides congruent t o the sides but is not similar to of EFGH, but rhombus IJKL Square EFGH, shown º is NOT congruent to square EFGH.B C F G J KMath 310 Ì SIM ILARITY cont ’d Ì 6. Triangle ABC has c ongruent sides AB & A C. The midp oint of BC is "E". Is AE.an alt itude of ªABC?What triangles are congruent, and how do w e know? A AB= AC, BE= CE are given. A E= AE (shared side) ... soªABE  ªACE by the “ SSS” congruen ce t heorem .What do we know about pABE and pACE?Since ªABE  ªACE, pABE and pACE must be congruent. B C What do we know about pAEB and pAEC? E Since ªABE  ªACE, pABE and pACE must be congruentSinc e they are su pplem ent ary, eac h m ust be 9 0°. 7. Given t hat: A B is t he midpo int of seg ment AD. C is t he midpo int of AF. E is the midpoint of segment DF. Wh at 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? D E 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 segmen ts A K and DK int o t hirds. B G J W hat can be sa id ab out segm ent s BJ and AK? They are PARALLEL! W hat can be sa id ab out tr iangles BJD an d A KD? Th ey are SIM ILA R.A F I K If A B is 5u, t hen, since B & C cut AD in thirds, BD must be 2@5u, i.e. 10u, and AD must be 15u.If BJ is 1 2 u, t hen A K mu st be 18u . 9. a. A b. E15D c.* B7 C 14 13 E D 5 1521 D 5 6 A C 9 B C A B BC | | ED AB || DE pA and pD are bot h right an gles. EAll segments that appear straight are straight (including ACD and BCE in 2nd and 3rd sketches).What triangles are similar an d w hy? Can y ou f ind t he missin g segm ent lengt hs?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) 10. A redwood tree casts a shadow 7 0' long at the same time a 10' tall street sign casts an 8' shadow.How tall must th e tree be?The sunlight streaming dow n, interrupted by t he vertic al tree and sign, f ormsthe shadow s on t he horizont al groun d. These right triangles have congruent baseangles sinc e the sunlight hits t he ground at t he same angle! By “ AA ” , t rianglesare similar. H 10 10' (70))) = )) û H = )))))))) = 87.5' 10'70' 8 8 70' 8' 6" 11. If held 10" from the wall, a spray gun paints a disc 3" in diameter. How fa r f rom th e w all sh ould t he g un be h eld to spr ay a t oy th at is 6 " w ide? 3"TOUGH 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 width of the spray is 3" wide at 10" distance, then it should be 6" wide when the spray gun is 20" away. 10"When t he w idth is doubled, t he area is quadrupled, so t he paint ends up th inner,by a f act or of 4. So the spray mu st be con tinued f our t imes as lon g— 4seconds.geo_sims7Math 310 Ì SIM ILA RIT Y so me proble ms Ì 12. The sun is low in the sky, and casts a sixteen-foot shadow f or a five-foot tall person. At t he same time, theshadow of the redw ood tree is 3 52 feet long. Wh at is t he heigh t o f t he redw ood tree?See #10 for reasoning. H 5'))) = ))) 352' 16' 5 352' H = ))) H? 16 = 5@22' = 110'5' 16' 352' 13. Tom , Billy and A nn liv e in a straight line. To m liv es four blocks n ort h and t w elve blocks w est of Billy. Billy liv essix blocks north of An n. Explain exact ly w here A nn liv es relativ e to Billy. T12Once you view the information in an appropriat e w ay (as in a m ap 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 on ly t hing w e don’ t k now is how many blocks w est of A nn Billy liv es.( ? ) 12 blocks))) = )))))))) 6 bl. 4 b l. AnnAnn is 18 b lock s east and 6 b lock s sout h of Bily. 14. * A 6' tall man standing 3' from a lamppost casts a shadow 4 .5' long. How long would his shadow be if he stood 6' away from the lamp?First set up the proportion for the initial situation given.Find …