Question 1: A tangent to a circle touches the circle at point P. If the radius to P is 5 cm and the length of the tangent from a point outside the circle is 13 cm, calculate the distance from that point to the center of the circle.

Answer: 12

Question 2: Construct a circle with radius 4 cm. Draw a tangent to the circle from a point 10 cm from the center. Label all relevant points.

Answer: Student's drawing

Question 3: In the diagram, the tangent touches the circle at point T. The radius to T is 6 cm. The angle between the radius and the tangent is 90°. Calculate the measure of the angle between two tangents drawn from the same external point if the distance between the points of tangency is 8 cm.

Answer: 90

Question 4: A circle has a radius of 7 cm. From a point outside the circle, the length of the tangent to the circle is 24 cm. Find the distance from the external point to the circle's center.

Answer: 25

Question 5: A tangent from point P touches a circle at point Q. If PQ = 10 cm and the radius QR = 6 cm, where R is the center of the circle, find the length of the segment from P to R.

Answer: 8

Question 6: A circle with radius 3 cm has a tangent from point A outside the circle. The distance from A to the circle's center is 10 cm. Calculate the length of the tangent from A to the circle.

Answer: 9.5

Question 7: In a diagram, two tangents are drawn from an external point to a circle, touching the circle at points T1 and T2. If the distance between T1 and T2 is 12 cm and the angle between the tangents is 60°, find the distance from the external point to the center of the circle.

Answer: 13

Question 8: A circle has a radius of 9 cm. A tangent is drawn from an external point, and the length of the tangent segment is 15 cm. Calculate the distance from the external point to the circle's center.

Answer: 17

Question 9: A circle with radius 5 cm is centered at O. From a point P outside the circle, a tangent is drawn to touch the circle at T. If OP = 13 cm, verify whether the length of PT (the tangent segment) is 12 cm. Show your working.

Answer: Yes, since 13^2 = 5^2 + 12^2

Question 10: Identify the common mistake in the following statement: 'The radius to the point of contact of a tangent is longer than the radius to any other point on the circle.' Correct the statement.

Answer: The radius to the point of contact of a tangent is always perpendicular to the tangent. It is not longer than other radii.