R is a point outside a circle and is 18 cm away from its centre. A secant drawn from the point R intersects the circle at points A and B in such a way that RA = 7 cm and AB = 5 cm. Find the radius of the circle in cm.

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Given:
- Distance from point R to center O = 18 cm
- RA = 7 cm
- AB = 5 cm

Since A and B lie on the circle, the secant from R intersects the circle at A and B.

Length RB = RA + AB = 7 + 5 = 12 cm

By the secant-tangent theorem (Power of a Point theorem):
RA * RB = (distance from R to center)^2 - (radius)^2

Let the radius be r.

Power of point R with respect to the circle is:
RA * RB = 7 * 12 = 84

Distance from R to center O is 18 cm, so:
RO^2 = 18^2 = 324

Using the power of point formula:
RA * RB = RO^2 - r^2
=> 84 = 324 - r^2
=> r^2 = 324 - 84 = 240

Therefore,
r = sqrt(240) = sqrt(16 * 15) = 4 * sqrt(15)

But this contradicts the options. Let's re-check the theorem used.

Actually, the power of a point theorem states:
For a point outside the circle, if a secant intersects the circle at points A and B, then:
RA * RB = (distance from R to center)^2 - r^2

Rearranged:
r^2 = RO^2 - RA * RB

Plugging in values:
r^2 = 18^2 - (7 * 12) = 324 - 84 = 240

r = sqrt(240) = 4 * sqrt(15)

So the radius is 4√15 cm.

Checking options again, option 1 is 4√15, which matches our calculation.

Hence, the correct answer is 4√15.

Answer

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