Points B and M are taken on a circle centered at point A, with BM = 10 cm. Point C is taken on the chord

Points B and M are taken on a circle centered at point A, with BM = 10 cm. Point C is taken on the chord BM so that BC = CM = CA. Find the radius of the circle.

Consider a triangle ABM.

Since BC = CM, then AC is the median.

Note that AB = AM, since they are the radii of the circle. Hence,

triangle ABM is isosceles. This means that the median AC is at the same time the height. It follows from this that the angle ACB is straight.

Consider now the triangle ABC. It is rectangular and isosceles since AC = BC.

We also know that BM = 10, which means BC = CM = AC = 10/2 = 5.

Then, by the Pythagorean theorem, we have:

AB ^ 2 = BC ^ 2 + AC ^ 2 = 5 ^ 2 + 5 ^ 2 = 2 * 5 ^ 2,

AB = 5 * √2

Answer: the radius is 5 * √2.