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.

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]