The radius of a circle can be determined with knowledge of its center coordinates and the point at which it is tangent. In this case, we have a circle with center (4,-5) that is tangent to the y-axis in the standard (x,y) coordinate plane.

Using basic geometry principles, we can calculate the radius of this circle by first noting that since it is tangent to the y-axis in our coordinate system, then the x value of its point(s) of contact must be equal to 0. Therefore, if we draw an imaginary line from (4,-5) to (0,-5), then these two points will make up one side of a right triangle whose hypotenuse will also mark out our desired radius.

To calculate this hypotenuse length exactly , let us use Pythagorean theorem r² = 4² + (-5)² . After some simple arithmetic rearranging equation substituting values yields 47 which equals our desired radius Thus it can seen easily attain exact measurements circles provided few guidelines followed correctly

It should also noted different methods calculation exist depending particular shape structure being discussed For example applying similar principles described above but for non centered circles involves calculating slopes determine respective radii However such calculations significantly more complex hence unsuitable beginners In conclusion finding radiuses circles relatively straightforward process involving basic math skills knowledge geometric properties

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