Question 1: In a circle with radius 7 cm, a tangent line touches the circle at point T. If a point P outside the circle is 15 cm from the center O, and PO = 15 cm, what is the length of the tangent segment PT?

Answer: 12

Question 2: Construct the tangent from a point 10 cm away from the center of a circle with radius 6 cm. Use the grid to accurately draw the tangent lines.

Answer: Constructed tangent segments

Question 3: A circle has a radius of 9 cm. A point Q outside the circle is 20 cm from the center O. Calculate the length of the tangent segment from Q to the circle.

Answer: 16

Question 4: A tangent line touches a circle with radius 5 cm at point R. If the distance from the external point S to the center O of the circle is 13 cm, find the length of the tangent segment SR.

Answer: 12

Question 5: A circle with radius 8 cm has a point outside it at a distance of 17 cm from the center. Calculate the length of the tangent from this point to the circle.

Answer: 15

Question 6: Explain why the tangent from an external point to a circle is always perpendicular to the radius at the point of contact.

Answer: Because the radius drawn to the point of contact is perpendicular to the tangent line, forming a right angle.

Question 7: In triangle OAT, where O is the center of the circle, AT is tangent at T, and OT is radius. If OT = 10 cm and the length of the tangent segment from point P outside the circle is 24 cm, find the distance OP.

Answer: 26

Question 8: A circle with radius 12 cm has an external point C such that the tangent segments from C to the circle are equal in length. If the point C is 20 cm from the circle's center, determine the length of each tangent segment.

Answer: 16

Question 9: A problem involves constructing two tangents to a circle from a point outside the circle. Using the grid, demonstrate the steps to accurately draw both tangent lines from a point 14 cm from the circle's center with radius 5 cm.

Answer: Constructed tangents

Question 10: Identify and correct the common mistake in the following statement: 'The tangent to a circle at the point of contact is always longer than the radius.'

Answer: The tangent is not longer; it is equal in length to the radius at the point of contact; in fact, the tangent from an external point is longer than the radius, but at the point of contact, the tangent and radius are perpendicular and do not compare in length as stated.