Mathematics

Question

Point C = (4/9,-5/9) lies on the circle with the center at S=(-2/3, 3/4). If CD is a diameter of circle S, find the coordinates for D. Answer must be given as a simplified fraction to receive full credit.

1 Answer

  • Answer:

    Step-by-step explanation:

    If CD is a diameter of circle S, then CD goes through circle S at point S.  CS is a radius, and so is DS.  That means that they are the same length.  That also means that S is the midpoint of CD.  We can use the midpoint formula and the 2 points we are given to find the other endpoint, D.

    [tex](-\frac{2}{3},\frac{3}{4})  =(\frac{\frac{4}{9}+x }{2},\frac{-\frac{5}{9}+y }{2})[/tex]

    To solve for x, we will use the x coordinate of the midpoint; likewise for y.  x first:

    [tex]-\frac{2}{3}=\frac{\frac{4}{9}+x }{2}[/tex]

    Multiply both sides by 2 to get rid of the lowermost 2 and get

    [tex]-\frac{4}{3}=\frac{4}{9}+x[/tex]

    Subtract 4/9 from both sides to get

    [tex]x=-\frac{16}{9}[/tex]

    Now y:

    [tex]\frac{3}{4}=\frac{-\frac{5}{9}+y }{2}[/tex]

    Again multiply both sides by that lower 2 to get

    [tex]\frac{3}{2}=-\frac{5}{9}+y[/tex]

    Add 5/9 to both sides to get

    [tex]y=\frac{37}{18}[/tex]

    And there you go!

    [tex]D(-\frac{16}{9},\frac{37}{18})[/tex]

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