find the coordinates of the points which are simultaneously 6 units from the points (6,6) and equidistant from the x and y axis. There are two ways to do this. I would like them both.

equidistant from the x,y axes means that if (x,y) is one point, y=x or y=-x

on the line y=x, if the distance from (6,6) is 6, then the x- and y-distances are ±6/√2.

So, (6-6/√2,6-6/√2) and (6+6/√2,6+6/√2) are two such points.

Points on the line y = -x are too far away from (6,6) to qualify.

I guess you could also do this by finding the intersection of

(x-6)^2 + (y-6)^2 = 36
and
y=x

the answers are the same.

To find the coordinates of the points that are simultaneously 6 units from the point (6,6) and equidistant from the x and y-axis, we can use two approaches: geometric and algebraic.

Approach 1: Geometric Method
1. Draw the point (6,6) on a coordinate plane.
2. Since the points of interest are equidistant from the x and y-axis, draw a line passing through the point (6,6) at a 45-degree angle (i.e., diagonally) to the coordinate axes.
3. Draw two circles with a radius of 6 units, one centered at (6,6) and another along the diagonal line drawn in step 2.
4. The intersection points of the circle centered at (6,6) and the diagonal line will be the desired points.

Approach 2: Algebraic Method
Let's find the coordinates of the points algebraically:
1. Consider a point (x,y) on the coordinate plane.
2. Using the distance formula, the distance between the point (x,y) and (6,6) is given by √((x-6)² + (y-6)²).
3. To be equidistant from the x and y-axis, the distance from the point (x,y) to the x-axis (y-coordinate) should be the same as the distance from the point (x,y) to the y-axis (x-coordinate).
4. Using the distance formula again, the distance from the point (x,y) to the x-axis is given by |y|, and the distance to the y-axis is |x|.
5. Setting up the equations √((x-6)² + (y-6)²) = |x| = |y|, we solve them simultaneously to get the coordinates of the desired points.

By following these two methods, you can find the coordinates of the points that are 6 units from (6,6) and equidistant from the x and y-axis.