Question 1: Given a circle with center O and a radius of 5 cm, a tangent touches the circle at point T. If OT is 5 cm, calculate the length of the segment from point T to a point P outside the circle such that TP is perpendicular to OT.

Answer: 3.6

Question 2: Construct a circle with radius 6 cm. From a point P outside the circle, draw a tangent to the circle. Measure and record the lengths of OP and PT, where O is the circle's center and T is the point of contact. Confirm that PT is perpendicular to the radius OT at T.

Answer: OP = 8.5 cm, PT = 6 cm, right angle at T

Question 3: A circle has a radius of 4 cm. A tangent from a point P outside the circle is 10 cm long. Find the distance from P to the center O of the circle.

Answer: 6.8

Question 4: Explain why the line from the point of contact to the circle's center is perpendicular to the tangent line. Use a diagram to illustrate your reasoning.

Answer: Because the tangent line at a point on a circle is perpendicular to the radius at that point, as proven by the properties of circle geometry.

Question 5: A circle with center O passes through point A. A tangent at point A intersects a line from P outside the circle. If OA = 7 cm and OP = 10 cm, find the length of the tangent segment from P to the circle.

Answer: 5.8

Question 6: In a circle, a radius of 8 cm is drawn to the point of tangency T. A point P outside the circle is 15 cm from the center O. Draw the tangent from P to the circle and calculate the length of the tangent segment PT.

Answer: 13

Question 7: A real-world scenario: A lighthouse is located 100 meters from a straight shoreline. The light beam from the lighthouse just touches a boat at point T on the water, forming a tangent line. If the lighthouse is 30 meters above sea level and the boat is on the water, find the shortest distance from the lighthouse to the boat, assuming the lighthouse, boat, and point T are in a straight line.

Answer: 70 meters

Question 8: Construct a circle with radius 9 cm. From a point P outside the circle, draw two tangents to the circle. If the distance from P to the circle's center is 15 cm, find the length of each tangent segment.

Answer: 12

Question 9: Identify the common mistake students might make when drawing a tangent from a point outside the circle. How can this mistake be corrected?

Answer: Drawing the tangent line at a wrong point; it should touch the circle at exactly one point and be perpendicular to the radius at that point.

Question 10: A circle with radius 10 cm has a tangent segment from point P outside the circle measuring 24 cm. Determine the distance from P to the center O of the circle.

Answer: 20

Question 11: Challenging Extension: Prove that the line from the point of tangency to the center of the circle is perpendicular to the tangent line using geometric theorems.

Answer: Using the definition of tangent and the property that a radius drawn to the point of tangency is perpendicular to the tangent line.