Proportions: A ratio is the quotient of two numbers. For example, 2 3

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1 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0) re in proportion, then we sy tht the numers re proportionl. Exmple, if =, then n re proportionl to n. Rememer from lger tht, if =, then =. The following proportions re equivlent: = = = = Theorem: Three or more prllel lines ivie trsversls into proportion. l m n In the piture ove, lines l, m, n n re prllel to eh other. This mens tht they ivie the two trnsversls into proportion. In other wors, = = = =

2 Theorem: In ny tringle, line tht is prlle to one of the sies ivies the other two sies in proportion. E F G In piture ove, F G is prllel to, so it ivies the other two sies of E, E n E into proportions. In prtiulr, = = = = Theorem: The isetor of n ngle in tringle ivies the opposite sie into segments tht re proportionl to the jent sie. In ove, is the isetor of. We hve: = = = =

3 Similr Tringles: Two tringles re si to e similr to eh other if they hve the sme shpe (not neessrily the sme size). If two tringles re similr, then ll three of their ngles re ongruent to eh other, n their orresponing sies re in proportion. This mens tht the rtio of their orresponing sies re equl to eh other. We use the symol to men similr to. So we write EF if tringle is similr tringle EF. In the piture elow,. This mens tht =, =, n =. Further more, their orresponing sies re in proportion, so = = = Theorem: If EF, n EF GHI, then GHI. In other wors, similrity is trnsitive. (ngle-ngle) Similrity Theorem: If two ngles of tringle is ongruent to two ngles of nother tringle, then the two tringles re similr. In piture ove, =, =, therefore, SS (Sie ngle Sie) Similrity Theorem: If n ngle is ongruent to n ngle of nother tringle the the sies tht mke up the ngles re in proportion, then the two tringles re similr.

4 In piture ove, if =, n, re in proportion with,, mening tht =, then SSS (Sie-Sie-Sie) Similrity: If ll three sies of tringle re in proportion with three sies of nother tringle, then the two tringles re similr. In piture ove, if the three sies of is proportionl to the three sies of, mening tht: = =, then Theorem: line prllel to one of the sies of tringle ivies the tringle into smller tringle similr to the originl one. E F G In piture ove, if F G, then F EG E Theorem: The ltitue of right tringle (through the right ngle) ivies the tringle into two similr right tringles tht re lso similr to the originl right

5 tringle. In piture ove, is right tringle with eing the right ngle. is the ltitue throught perpeniulr to. We will prove tht Proof: Sttements Resons 1. is ltitue to 1. given ef. of ltitue 3. n re right s 3. ef. of lines 4. m + m = ngle ition Postulte 5. omplementry to 5. ef. of omp. ngles 6. m + m = Sum of Interior ngles of 7. omplementry to 7. ef. of omp. ngles 8. = 8. s omp. to sme re = 9. m + m = Sum of Interior ngles of 10. omplementry to 10. ef. of omp. ngles 11. = 11. s omp. to sme re = The previous theorem out the ltitue of right tringle iviing the right tringle into three similr tringles llows us to prove one of the most importnt theorem in geometry (n mthemtis in generl): Pythgoren Theorem: In ny right tringle, the sum of the squre of the length of the two legs is equl to the squre of the length of the hypotenuse.

6 h e h h e To prove the pythgoren theorem, note from the previous theorem tht. Therefore, their orresponing sies re in proportion. In prtiulr, = ; = e The first eqution gives us: = = 2 The seon eqution gives us: = e e = 2 Notie tht e + =, therefore, we hve: e + = = = = 2

7 Exmple: 5 7 h 2 6 In first tringle, = = 49 2 = 24 = 24 = 2 6 In seon tringle, (2) 2 = h 2 h 2 = = 10 h = 10 Speil Tringles: right tringle where one of the ute ngle is 30 n the other ute ngle is 60 is lle tringle. Theorem: In tringle, if the sie opposite the 30 ngle hs length of, then the sie opposite the 60 ngle hs length of 3, n the hypotenuse hs length In, is the sie opposite the 30 ngle, if length of =, then the sie opposite the 60 ngle, sie, will hve length of 3, n the hypotenuse hs length 2. We n prove the tringle y rwing the ltitue to the equilterl tringle s ove. Let e the length of eh of its sie. Notie tht the ltitue ivies the eqilterl tringle into two ongruent tringles (why?). Eh of the tringle is tringle n the length of is hlf of. Using Pythgoren theorem, we see tht 2! 2 + h 2 = h2 = 2 h 2 = 2 2 h 2 = h2 = 32 4 h = 4 3 = 3 2 2

8 30 h 60 /2 / tringle is n isoeles right tringle. In n isoeles right tringle, eh ute ngle is the sme, therefore, eh ute ngle is 45. Theorem: In tringle, if the length of one of the leg is, then the length of the other sie is lso, n the length of the hypotenuse is The tringle n e prove just using the Pythgoren theorem. In, if one of the leg hs length, the the Isoeles tringle theorem tells us the other leg must lso hs length sine the se ngles re ongruent. Using the Pythgoren theorem, the length of the hypotenuse, h, is: h 2 = = 2 2 h = 2

9 If two hors interset insie irle, the prout of the lengths of the segments of one hor is equl to the prout of the lengths of the segments of the other. E In the irle ove,, re hors interset t E, the theorem tells us tht: E E = E E If two sent lines interset outsie irle, the prout of the length of one of the sents with the length outsie the irle is equl to the prout of the length of the other sent with the length outsie the irle. E In irle ove, sents, interset the irle t points n E, respetively, n we hve: = E If tngent n sent to irle interset outsie the irle, the squre of the length of the tngent is equl to the prout of the length of the sent times the length of the sent outsie the irle.

10 In irle ove, tngent intersets outsie the irle t, we hve: =