The chord of the circle is equal to a and subtracts the arc by 90 degrees. Find the radius of the circle.

From the center of the circle O we draw the radii of the circle OB and OA, then the triangle AOB is isosceles. Since the chord AB contracts the arc at 90, the central angle resting on this chord is also equal to 90. Then the triangle AOB is isosceles and right-angled.

From the right-angled triangle AOB, by the Pythagorean theorem, we determine the lengths of the legs OA and OB.

OA ^ 2 + OB ^ 2 = AB ^ 2.

2 * OA ^ 2 = a ^ 2.

OA ^ 2 = a ^ 2/2.

ОА = R = a / √2 = a * √2 / 2 cm.

Answer: The radius of the circle is a * √2 / 2 cm.