Question 1: In a circle with radius 7 cm, a tangent touches the circle at point T. If the distance from the center to the point where the tangent intersects a secant line outside the circle is 15 cm, find the length of the tangent segment from the point outside the circle to point T.

Answer: 12

Question 2: Construct a circle with radius 5 cm. Draw a tangent from a point 8 cm from the circle's center. Measure and record the length of the tangent segment.

Answer: See full answer key

Question 3: A circle has a radius of 9 cm. A point outside the circle is located 20 cm from the center. Draw the tangent from this point to the circle and determine the length of the tangent segment.

Answer: 16

Question 4: True or False: The length of a tangent from an external point to a circle is always equal to the radius of the circle.

Answer: False

Question 5: If two tangents are drawn from an external point to a circle, what is the relationship between the lengths of these two tangent segments?

Answer: They are equal

Question 6: Calculate the length of the tangent segment if the external point is 25 cm away from the circle's center and the radius of the circle is 7 cm.

Answer: 24

Question 7: A circle with radius 6 cm has a point outside the circle. The distance from this point to the center is 10 cm. Draw the tangent from this point and find its length.

Answer: 8

Question 8: Identify the common mistake: A student claims that the tangent length from an external point is always equal to the circle's radius. Is this correct? Explain briefly.

Answer: No, because the tangent length depends on the distance from the external point to the circle, not just the radius.

Question 9: In triangle OPA, where O is the center of the circle, P is the external point, and A is the point of tangency, which angles are equal? Explain your reasoning.

Answer: Angle OPA equals angle OAP because the tangent at A is perpendicular to the radius OA, forming right angles and congruent angles in the triangle.

Question 10: A tangent is drawn to a circle with radius 10 cm from a point outside the circle. If the tangent length is 24 cm, what is the distance of the external point from the circle's center?

Answer: 26

Question 11: Construct a circle with radius 4 cm. From a point outside the circle, draw two tangents. Measure and compare the lengths of the tangents. Are they equal? Why?

Answer: Yes, because the tangents from a common external point are equal in length.