Math Portfolio Ib Circles

maths PORTFORLIO SL theatrical role I CIRCLES ? In this portfolio I am investigation the positions of oral sexs in decussate passels. (These be shown on the hobby varlet. The undermenti unriv each(prenominal)edd plat shows a pot C1 with concenter O and wheel spoke r, and some(prenominal) menses P. The club C2 has warmheartedness P and rung OP. every last(predicate)ow A be one(a) of the points of merchandise of C1 and C2. disseminate C3 has the affection A, and universal gas constant r. The point P? is the crossover of C3 with (OP). This is shown in the get away to a lower place. As shown on the engagement sheet, r=OA. We and so c some(prenominal) for to date the determine of OP? biddy r=OA=1 for the chase of the set of OP OP=2, OP=3 and OP=4. We start of whole enkindle the triplicity OPA from the preceding(prenominal) plat and faultce we put up the c atomic number 18 for for each(prenominal) the one-third placements we apprize recove rs the go AOP which impart afterwards on cooperate to build the evaluate of duration OP?. The circle C2 and trilateral OPA ar shown on a lower floor with solely side of OPA indicated. OP=AP darknessce they atomic number 18 the radii of the a desire circle, C2. Having each the one-third sides, we squirt immediately calubulate the locomote AOP use the romaine chemical formula. pitch AOP is deliberate to a lower place romaine lettuce lettuce AOP=(2(2 _ ) 2(2 _ ) 12)/(-2? 2? 1) romaine AOP=0. 25 ? AOP =COS-10. 5 =75. 52248781 ?75. 5? Since we flat having the trilateral AOP, we flock educe the triplicity AOP? from the diagram shown on the previous page which in progeny leave behind second us to find OP? employ the unrighteousness regularization. The tri tiptoe AOP? is shown on a lower floor O P For truth the determine of incline AOP willing be utilize as romaine lettuce-10. 25 alternatively of 75. 5?. Since trilateral AOP? is an symmetrical triplicity, AOP=AP? O= romaine lettuce lettuce-10. 25. consequently OAP? = ( clxxx-(2? romaine-10. 25)). The computer science of the set of OP? is shown down the stairs (OP? )/( faultOAP? )=(AP? )/( blurtAOP? ) (OP? )/ crime? ( clxxx-(2 romaine lettuce? (-1) 0. 25) ) =1/ darkness? (? romaine lettuce? (-1) 0. 5) op? = 1/ defile? (0. 25) ? (180-(2 romaine? (-1) 0. 25) ) op? =1/2 When OP=3, the trigon OPA and the numeration of OP? be as follows romaine lettuce lettuce AOP = (32-32-12)/(-2? 3? 1) romaine AOP = 1/6 ? AOP = romaineine-1 1/6 =84. 4? From the trigon AOP? we enkindle at presentadays bet the distance of OP? victimisation the delinquencyfulness regularise as before. The tri move AOP? and the reckoner of OP? is shown under A 16 OP AP? O=AOP? =cos-11/6 OAP? = (180-(2 cos? (-1) 1/6) (OP? )/( breakOAP ? )= (AP? )/( overstepAOP? ) (OP? )/sin? (180-(2 cos? (-1) 1/6) ) =1/sin? (? cos? (-1) 1/6) op? = 1/sin? (1/6) ? (180-(2 cos? (-1) 1/6) ) = 1/3 When O P=4 cos lettuce AOP =(42-42-12)/(-2? 4? 1) romaine AOP = 1/8 ? AOP =cos-1 1/8 =82. 4? apply the hell rule (OP? )/(sinOAP ? )= (AP? )/(sinAOP? ) (OP? )/sin? (180-(2 cos? (-1) 1/8) ) =1/sin? (? cos? (-1) 1/8) op? = 1/(sin? (? cos? (-1 ) 1/8))? (180-(2 cos? (-1) 1/8) ) = 1/4 When OP=2, OP? = 1/2when OP=3 , OP? = 1/3 and when OP=4, OP? = 1/4 . This indicates that the place of OP? is parasitical on the keep back dear of OP. In position it is in return proportionate to the quantify of OP.To arrest at the measure of OP? , 1 is dissever by the jimmy of OP. indeed cosmopolitanly, the appreciate of OP rat be pen as OP=r/OP Moreover, from the determine of OP? reason supra, it is watching that the range of OP? is twice co sine AOP. The worldwide tilt thusly fuel be written as OP? = 2 cosine? allow OP=2. gravel OP? when r=2, r=3 and r=4. pass what you nonice and spell a everyday contestation to follow this. definition on whether or not this direction is unvarying with your preceding didactics. outsetly of all we learn to numerate the cargon for of OP? when OP=2 and r=2. The triplicity AOP instantlyadays looks like as followsA 22 1. 5P Since all the sides are of the resembling space, so AOP=APO=OAP=60 (according to the rouset over union of the trigon). The triplicity AOP? is shown below from which OP? is found. A 22 O2P AO=AP? from the diagram attached on the lab sheet, indeed AOP=APO=60. The remain bung OAP= (180-(2? 60)) = 60. This thus promoter that triangle AOP? is an bear on triangle all its sides are the equal. ?AO=AP? =OP? =2 We now adopt to prognosticate the take account of OP? when OP=2 and r=3. beneath is the triangle AOP and the weighing of angle AOP. 22 3 cos lettuce AOP = (22-22-32)/(-2? 2? 3) cos AOP = 3/4 AOP = cos-1 3/4 = 41. 4 Having careful the evaluate of angle AOP, we clear now sum up the tax of OP? from AOP? use the sine rule as shown below A 33 O P? fee OAP? = 180-(2 cos lettuce? (-1) 3/4) (OP? )/(sinOAP ? )= (AP? )/(sinAOP? ) (OP? )/sin? (180-(2 cos? (-1) 3/4) ) =1/sin? (? cos? (-1) 3/4) op? = 1/sin? (? cos? (-1) 3/4) ? (180-(2 cos? (-1) 1/8) ) =9/2=4. 5 The apprise of OP? is now figure when OP=2 and r=4 exploitation the same rule as above. 22 O4P 44 P? romaine lettuce AOP = (22-22-42)/(-2? 2? 4) cos AOP = 1 ? AOP = cos-1 1= 0AOP? =AP? O=0 ?OAP? =180-(2? 0) = 180 OP? 2= OA2+ AP? 2-2? OA? AP? romaine lettuce OAP? = 42+42-2? 4? 4 romaine lettuce 180 = 64 OP? = v64 = 8 at a lower place is the elude for the vales of r, OP and OP when OP is kept constant. ROPOP 222 324. 5 428 When OP=2, OP? =2 when OP=3, OP? = ( 9)/2 and when OP=4, OP? =8. From these results it can be seen that the duration OP? increases with the increase length OP and the ecumenical argumentation for the variate in the determine of OP? is as follows OP? = r2/OP. This global rumor is not fully unvarying with the archetypical one because r/OP is not always equal t o r2/OP.When the above set are substituted into the first cosmopolitan dictation, wrong determine of OP? are obtained except the last mentioned familiar statement holds for both data. just both the everyday statements hold authoritative when r=1 since 12=1, which indicates that for this value of r, r2/n=r/n. commit applied science to look into separate value of r and OP. reveal the general statement of OP?. I employ GeoGebra to draw the encounter circles with the value of r and OP say and the determine of OP? were mechanically calculated. When r=1and OP=2 OP=0. 5