In the circle, the radius of which is equal to 5, the chord AB = 8. Point C lies on the chord AB so that AC: BC = 1: 2.

| May 21, 2021 | education

In the circle, the radius of which is equal to 5, the chord AB = 8. Point C lies on the chord AB so that AC: BC = 1: 2. Find the radius of the circle tangent to the given circle and tangent to the chord AB at point C.

Determine the distance from the center of the circle to the chord AB. From a right-angled triangle OBM

ОМ ^ 2 = ОВ ^ 2 – ВМ ^ 2 = 25 – 16 = 9. ОМ = 3 cm.

Point C divides the chord AB into segments AC and BC. By condition, BC = 2 * AC, then AC + 2 * AC = AB = 8 cm. AC = 8/3 cm. AM = AB / 2 = 8/2 = 4 cm.

Then CM = OH = AM – AC = 4 – 8/3 = 4/3 cm.

Let the radius of the small circle be equal to X cm, O1C = O1P = X cm.

In a right-angled triangle OO1H, HO1 = (O1C + CH) = (X + 3), OO1 = (OK – O1K) = (5 – X).

Then, by the Pythagorean theorem:

(5 – X) ^ 2 = (X +3) ^ 2 + (4/3) ^ 2.

25 – 10 * X + X ^ 2 = X ^ 2 + 6 * X + 9 + 16/9.

16 * X = 16 – 16/9.

X = 1 – 1/9 = 8/9 cm.

Answer: The radius of the smaller circle is 8/9 cm.



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