Question 1: In circle O, a radius is drawn to point A on the circle. A tangent at A is drawn. If the radius AO is 6 cm, what is the length of the segment from the center O to the point where the tangent intersects the circle?

Answer: 6 cm

Question 2: Construct a circle with center O and radius 4 cm. Draw a radius to point A on the circle and then draw the tangent at A. Verify that the tangent is perpendicular to the radius.

Answer: Perpendicular

Question 3: Calculate the angle between the radius and the tangent at point A if the radius is 5 cm and the tangent touches the circle at A. (Hint: use the property that the tangent is perpendicular to the radius).

Answer: 90°

Question 4: A circle has radius 7 cm. A tangent at point A forms a 30° angle with the line from the center to point A. What is the length of the tangent segment from point A to the point where it meets the tangent line?

Answer: 14√3 cm

Question 5: Explain why the tangent to a circle at point A is perpendicular to the radius drawn to A. Include a brief geometric proof or reasoning.

Answer: Because the tangent line only touches the circle at A and the radius drawn to A is perpendicular to the tangent at the point of contact, by the tangent-secant theorem or geometric reasoning.

Question 6: On a circle with radius 10 cm, point A is on the circle. The radius AO is drawn, and a tangent at A is constructed. If the tangent is 8 cm long from A to the point where it intersects a line outside the circle, find the distance from the center O to this external point.

Answer: 10 cm

Question 7: Construct a circle with radius 3 cm. From an external point P outside the circle, draw a tangent to the circle. Measure the distance from P to the center O of the circle.

Answer: Greater than 3 cm

Question 8: A circle has radius 9 cm. A tangent at point A makes a 45° angle with the line from the center to A. Calculate the length of the tangent segment from A to its intersection point with the tangent line.

Answer: 9√2 cm

Question 9: A student claims that the tangent to a circle at point A is always parallel to the radius AO. Is this correct? Explain your reasoning.

Answer: No, because the tangent is perpendicular to the radius at A, not parallel.

Question 10: Identify and correct the mistake in the following statement: 'The tangent to a circle is always at a 45° angle to the radius.'

Answer: The tangent is always perpendicular (90°) to the radius, not 45°.