Question 1: Define a tangent to a circle and state one key property of a tangent line in relation to the circle.

Answer: A tangent touches a circle at exactly one point and is perpendicular to the radius at that point.

Question 2: Calculate the length of the tangent from an external point 10 cm away from the center of a circle with radius 6 cm.

Answer: 8

Question 3: In a circle, a tangent line touches the circle at point P. If the radius OP is 7 cm, what is the length of the tangent segment from an external point Q to P, given that Q is 15 cm from O?

Answer: 13

Question 4: Plot a circle with radius 5 cm and draw a tangent at a point on the circle. Label all relevant points.

Answer: Student draws circle, tangent at point, labels radius, tangent point, external point.

Question 5: A circle has a radius of 9 cm. An external point is 20 cm from the center. Find the length of the tangent from this point to the circle.

Answer: 16

Question 6: Explain why the radius drawn to the point of contact of a tangent is perpendicular to the tangent line.

Answer: Because the tangent line touches the circle at one point and the radius at that point is perpendicular to the tangent property, ensuring the tangent only touches once and the radius is perpendicular.

Question 7: A circle with radius 12 cm has a point outside it such that the tangent segments to the circle from this point are both 13 cm long. Find the distance from the external point to the circle's center.

Answer: 5

Question 8: Construct a triangle with two sides 8 cm and 15 cm meeting at a point outside a circle of radius 10 cm, such that the two lines are tangents. Show all steps.

Answer: Student constructs two tangent lines, labels points, shows equal tangents, and marks the circle.

Question 9: Identify and correct the mistake in the following statement: 'The tangent to a circle is always parallel to the radius at the point of contact.'

Answer: The tangent is perpendicular, not parallel, to the radius at the point of contact.

Question 10: A real-world scenario: A lighthouse stands 40 meters from the cliff. The beam of light from the lighthouse just touches a boat floating at sea, 50 meters from the lighthouse along the line of sight. How far is the boat from the cliff?

Answer: 30