The circle with center C shown above is tangent to both axes…
We begin by creating a right triangle with the x-axis, the radius of circle C to the x-axis and the line segment from center of circle C to the origin. We see that we have created an isosceles right triangle, also known as a 45-45-90 degree right triangle. We know this because each leg of the right triangle is equal to a radius of the circle. We can label all this in our diagram.
We know the side-hypotenuse ratios in a 45-45-90 degree right triangle are:
x:x:x√2, where x represents the leg of the triangle and x√2 is the hypotenuse.
We can use this to determine the leg of the triangle.
Since x√2 equals the hypotenuse of the triangle we can say:
x√2 = k
x = k/√2
Since x also represents the radius of the circle, k/√2 is equal to the radius.