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By Zoque E.

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Fig. 3-20. CHAPTER 3 The xy Coordinate Plane 43 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a ¼ ðÀ16 À ðÀ21ÞÞ2 þ ðÀ8 À 4Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðÀ16 þ 21Þ2 þ ðÀ12Þ2 ¼ 25 þ 144 ¼ 13 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ¼ ðÀ4 À ðÀ16ÞÞ2 þ ðÀ3 À ðÀ8ÞÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðÀ4 þ 16Þ2 þ ðÀ3 þ 8Þ2 ¼ 144 þ 25 ¼ 13 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ ðÀ9 À ðÀ4ÞÞ2 þ ð9 À ðÀ3ÞÞ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðÀ9 þ 4Þ2 þ ð9 þ 3Þ2 ¼ 25 þ 144 ¼ 13 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ ðÀ21 À ðÀ9ÞÞ2 þ ð4 À 9Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðÀ21 þ 9Þ2 þ ðÀ5Þ2 ¼ 144 þ 25 ¼ 13 Because a ¼ b ¼ c ¼ dó ðÀ21ó 4Þó ðÀ4ó À3Þó ðÀ9ó 9Þ and ðÀ16ó À8Þ are the vertices of a square.

1. 2. 3. 4. 1 3 jx þ 4j À 5 < 2 4 þ 2j3x þ 5j ! 6 9 þ j6 À 2xj 11 À3j7x þ 4j À 4 < À13 SOLUTIONS 1. 1 jx þ 4j À 5 < 2 3 1 jx þ 4j < 7  3  1 3 jx þ 4j < 3ð7Þ 3 jx þ 4j < 21 À21 < x þ 4 < 21 À25 < x < 17 ðÀ25ó 17Þ 2. 4 þ 2j3x þ 5j ! 6 2j3x þ 5j ! 2 2j3x þ 5j 2 ! 2 2 j3x þ 5j ! 1 CHAPTER 2 Absolute Value 3x þ 5 À1 3x À6 x À2 27 3x þ 5 ! 1 3x ! À 3  4 ðÀ1ó À 2Š [ À ó 1 3 3. 9 þ j6 À 2xj j6 À 2xj À2 6 À 2x À8 À2x 11 2 2 À4 À8 À2 À4 ! x! À2 À2 À2 4 ! x ! 2 or 2 x 4 ½2ó 4Š 4. À3j7x þ 4j À 4 < À13 À3j7x þ 4j < À9 À3j7x þ 4j À9 > À3 À3 j7x þ 4j > 3 7x þ 4 < À3 7x < À7 x < À1 7x þ 4 > 3 7x > À1 1 x>À 7   1 ðÀ1ó À 1Þ [ À ó 1 7 Chapter 2 Review 1.

We could use either ð2ó 1Þ or ðÀ4ó 9Þ in the equation to find r2 . Here, ð2ó 1Þ will be used. ð2 þ 1Þ2 þ ð1 À 5Þ2 ¼ r2 9 þ 16 ¼ r2 25 ¼ r2 The equation is ðx þ 1Þ2 þ ðy À 5Þ2 ¼ 25. 2. The midpoint will be the center of the circle.     0 þ ðÀ12Þ 4 þ 9 13 ðhó kÞ ¼ ó ¼ À6ó 2 2 2 2 2 So far, we know the equation is ðx þ 6Þ2 þ ðy À 13 2 Þ ¼ r . We will use 2 ð0ó 4Þ to find r .   13 2 ð0 þ 6Þ þ 4 À ¼ r2 2  2 5 2 6 þ À ¼ r2 2 25 36 þ ¼ r2 4 169 ¼ r2 4 2 169 The equation is ðx þ 6Þ2 þ ðy À 13 2Þ ¼ 4 .

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A basis for the non-crossing partition lattice top homology by Zoque E.


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