In a circle with center O and radius 12, the point M is the midpoint of radius OA, a chord BC = 21

In a circle with center O and radius 12, the point M is the midpoint of radius OA, a chord BC = 21 is drawn through M. Find BM and MC.

Let us drop the perpendicular from the center of the circle to the chord and connect the center of the circle with points A and B.

COB – isosceles triangle, OK – its height.

Find the square of the leg OK of the right-angled triangle OBK:

OK ^ 2 = OB ^ 2 – (BC / 2) ^ 2;

By the condition ОМ = ОА / 2 = 12/2 = 6.

Let’s find the leg KM of the right-angled triangle OMK:

MK = √ (OA / 2) ^ 2 – OK ^ 2) =

= √ ((OA / 2) ^ 2 – OB ^ 2 + (BC / 2) ^ 2) = √ (6 ^ 2 -12 ^ 2 + 10.5 ^ 2) = 1.5;

BM = BC / 2 – MK = 21/2 – 1.5 = 9;

КС = ВС – ВМ = 21 – 9 = 12.

Answer: BM = 9, KС = 12.