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# Intermediate Russian RUSSIAN 4

UCLA

GPA 3.96

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This 3 page Class Notes was uploaded by Valentine Witting on Friday September 4, 2015. The Class Notes belongs to RUSSIAN 4 at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 102 views. For similar materials see /class/177887/russian-4-university-of-california-los-angeles in Russian at University of California - Los Angeles.

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Date Created: 09/04/15

Point Mass Geometry part ll October 317 2008 In each of the problems below the key is to put some masses at some of the points in the problem More problems 1 The base ABCD of a pyramid FABCD is a parallelogram The plane a intersects AF7 BF7 CF and DF at points A17B17C17D1 respectively Given that AA BB CC 1 1 1 2 5 10 lA1Fl lB1Fl lC1Fl lDDll nd the ratio 1 DIF 2 Let L 6 AC and M 6 BC be the points on the sides of AABC so that lCLlalCAl lCMl lCBl 0lta lt1i Let P AM n BL Find the ratio Center of Mass Via vectors 0 Let Z ZmlA7 mgB Then the condition mldl 721ng can be written as 7211127111 7212127121 Since ZA1 and Z142 have opposite directions7 it follows that m1ZA1 m2ZA2 or o This condition can be taken as the de nition of the center of mass of the system of two points In the case when there are more then two masses7 we get the following The center of mass of the system ofn point masses m1A17m2A2i i imnAn is the point Z such that 7211271 m2ZA2 1 H ngAn 0i i Show that if Z ZmlAlmgAg7 then for any point 0 we have W mlOAl mgOA2 m1 m2 Is it true that if for any point 0 and a point Z the equality above holds7 then Z is the center of mass of A1 and A27 0 i Let G be the point of intersection of medians of AABC Find E in terms of E and A i 9 Let C be the point on a segment AB such that lACl lCBl 2 z 7 Expess this fact using the notion of the center of mass Express the same fact using vectors t Let G and G be the centers of mass of AABC and AAlBlCi respectively Show that 0801 gm 371 Tel Negative masses The same de nition of the center of mass works if some or all of the masses in the system are negative 0 Let m1A1 and mgAg be two point massesi Assuming that m1 m2 y 07 the center of mass of this system Z Zm1A1mgA2 lies on the line A1142 so that lmildi lm2l d27 where d1 lZAll and d2 lZAgl The center of mass Z lies between A1 and A2 if and only if the masses have the same sign i e 7 both are positive or both are negative i Show that if ABCD is a parallelogram7 then mD ZmA7 7mB7 721639 N The base of a pyramid SABCD is a parallelogram ABCD A plane a intersects the sides SA7 SBSC and SD at points A17B17C17D1 respec tively Given that lSAll lSAL lSBll lSBl and lSC ilSCL nd the ratio lSDlllSDl use the previous problem Barycentric coordinates 0 Let M be point inside of AABC One can nd masses m17m27m3 so that M ZmlA7 mgB7 mgC o The masses m1 m2 7213 are de ned up to a constant factor ZkmlA7 kmgB7 kmgC Zm1A7m2B7 mac 0 De ne k so that the sum of masses equals to 1 For any point M inside of AABC there are positive numbers M1M2h3 so that 0 M1M2M319 M ZM1A7 237 30 These numbers are called the baricentric coordinates or Bcoardinates of M with respect to AABCi Warm up problems 1 Find the baricentric coordinates of each of the vertices 2 Find the point whose baricentric coordinates with respect to AABC are equal to 12 12 0 3 Find the baricentric coordinates of the point of intersection of medians 4 Let M1M2M3 be the baricentric coordinate of a point M with respect to AABC Prove the following statements a M lies on the line trough AB if and only if 3 0 b M lies on the segment AB if and only if M1t2 gt 0 and 3 0 5 Draw a triangle AABC and mark the following points with given bari centric coordinates M12 14 147 M7120 6 Let M 6 BC be such that lBMl 13lBCl Let N 6 AB be such that lANl lABl Find the baricentric coordinates of the point of intersection of AM and ON Problems 1 Show that the numbers M1M2M3 satisfying conditions in the de nition of baricentric coordinates exist and are unique for any point M on the plane Hint Use the fact that M is the center of mass of A B C if and only if for any point P we have W mm Mfg Mail Choose point 0 in a smart way to prove the statement 0 Based on the previous problem describe a graphical way to determine the baricentric coordinates of any point M with respect to a given triangle AABC 9 Find the baricentric coordinates of the point of intersection of altitudes of acute angle triangle AABC given that its sides have lengths a 25 and the angles are equal to 157

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