Question 1: In a circle with center O, a radius OB is drawn to point B on the circle. A tangent at B touches the circle at point B. What is the measure of the angle between OB and the tangent at B?

Answer: 90 degrees

Question 2: Construct a circle with radius 4 cm. Draw a radius and a tangent at the point of contact. Verify that the radius is perpendicular to the tangent.

Answer: Constructed with right angles at the point of contact

Question 3: A circle has radius 5 cm. A point P outside the circle is 12 cm from the center. Calculate the length of the tangent from P to the circle.

Answer: 13 cm

Question 4: Explain why the radius drawn to the point of contact of a tangent is perpendicular to the tangent.

Answer: Because the radius and tangent form a right angle at the point of contact, as per the circle theorem.

Question 5: Given a circle with center O, a radius OB, and a tangent at B, if the radius OB is extended past B, what is the measure of angles formed between the extended radius and the tangent?

Answer: 90 degrees

Question 6: A tangent is drawn to a circle at point C. If the radius to C is 6 cm, and the tangent touches the circle at C, what is the distance from point C to the center of the circle?

Answer: 6 cm

Question 7: A circle has a radius of 3 cm. A tangent from an external point P is 8 cm from the center. Find the length of the tangent from P to the circle.

Answer: ~7.62 cm

Question 8: A diagram shows a circle with radius 7 cm. A tangent at point D touches the circle, and a radius OD is drawn to D. If the tangent is extended, what is the relationship between the radius OD and the tangent at D?

Answer: They are perpendicular (90 degrees) at the point of contact.

Question 9: Identify the common mistake students make when drawing a radius and tangent, and state how to correct it.

Answer: Mistake: Drawing the radius not perpendicular to the tangent. Correction: Ensure the radius is drawn at a right angle to the tangent at the point of contact.

Question 10: Challenge: In a circle, a tangent at point E is extended, and the radius OE makes an angle of 85° with the tangent. Is this possible? Explain.

Answer: No, because the radius should be perpendicular (90°) to the tangent at the point of contact; 85° indicates an error.