SOLUTION: find all points having an x coordinate of 9, whose distance from the point (3,-2) is 10

All points which have x coordinate 9 are on the vertical
green line below:



we take a compass and open it up to a radius of 10 units.
We put the sharp point of the compass on the point (3,-2) 
and we find that can swing two red arc to cut the green vertical
line in two points, like this:



Then we draw 10-unit long blue lines from (3,-2) to the 
points where the arcs cut the green line:



Those two points look like they are (9,6)  and (9,-10), but
looking and estimating is not good enough for mathematics.
We must calculate them:

Let's draw a black horizontal line segment from (3,-2) over to the
green vertical line:



That line segment is 6 units long because it extends horizontally
from the point (3,-2) over to (9,-2) and that is 6 units.  

 

If we look just at the top half, we have a right triangle, with
hypotenuse 10 units and horizontal leg 6 units, like this.  Call the
length of the green side of the right triangle h,



By the Pythagorean theorem








So the upper vertex of that right triangle
is 8 units above (9,-2) so it must be (9,6),
just as we guessed just by looking.

Putting back the lower right triangle, which is congruent to
the upper one:



So the lower vertex of that lower right triangle
is 8 units below (9,-2) so it must be (9,-10),
just as we guessed just by looking.



Edwin