In the figure, line l touches the circle with center O at point P.

Explanation:- 

We are given a circle with radius r and a tangent line l that intersects the circle at point P. We are also given a chord RS that is parallel to line l. We need to find the radius of the circle.

We start by using the tangent theorem, which tells us that a line tangent to a circle is perpendicular to the radius at the point of contact. Therefore, we have seg OP perpendicular to line l.

Since chord RS is parallel to line l, we can use the perpendicular bisector theorem to find that QS = 1/2 RS. This is because the perpendicular drawn from the center of the circle to a chord bisects the chord.

We are also given that RS = 12 cm, so QS = 1/2 * 12 = 6 cm. Therefore, QR = QS = 6 cm.

Next, we use the fact that Q is the midpoint of OP to find that OQ = 1/2 OP. Since we know that seg OP has length r, we have OQ = 1/2 r.

Now, we can use the Pythagorean theorem in triangle OQS to relate the sides OQ, QS, and OS. Since seg OP is perpendicular to chord RS, angle OQS is a right angle. Therefore, we have:

OS^2 = OQ^2 + QS^2

Substituting OQ = 1/2 r and QS = 6, we get:

r^2/4 = OS^2 – 36

We also know that the radius of the circle is r, so we can substitute r^2 for OS^2 to get:

r^2/4 = r^2 – 36

Multiplying both sides by 4 gives:

r^2 = 4r^2 – 144

Simplifying, we get:

3r^2 = 144

Dividing both sides by 3 gives:

r^2 = 48

Taking the square root of both sides, we get:

r = 4 sqrt(3)

Therefore, the radius of the circle is 4 sqrt(3) cm.