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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