Question 1: A tangent to a circle is a line that touches the circle at exactly one point. The tangent line is perpendicular to the radius drawn to the point of contact.

Answer: Perpendicular to radius at point of contact

Question 1: Calculate the length of the tangent segment from a point 10 cm away from the circle's center to the point of contact, given the radius is 6 cm.

Answer: 8 cm

Question 2: If a tangent is drawn at a point on a circle with radius 5 cm, and the tangent segment from an external point is 12 cm, what is the distance from the external point to the circle's center?

Answer: 13 cm

Question 3: Identify whether the following statement is true or false: 'A tangent line can intersect a circle at two points.'

Answer: False

Question 1: A circle has a radius of 7 cm. A point outside the circle is 15 cm from the center. Draw the two possible tangent lines from this point to the circle. Explain why both tangents are equal in length.

Answer: Tangent segments are equal because they are both from the same external point to the circle, and tangent segments from a common external point are equal in length.

Question 2: Prove that the tangent at a point on a circle is perpendicular to the radius at that point.

Answer: Using the properties of a circle and the fact that the radius to the point of contact is perpendicular to the tangent, which can be proved using congruent triangles.

Question 1: A lamp post is 8 meters tall. A viewer standing 10 meters from the lamp post observes a shadow where the tangent from the top of the lamp to the tip of the shadow forms a right angle with the ground. Find the length of the shadow.

Answer: 6.4 meters

Question 1: Given a circle with center O and radius 9 cm, and a point P outside the circle such that OP = 15 cm. The two tangents from P touch the circle at points A and B. Find the length of the tangent segments and the angle between the two tangents.

Answer: Tangent length = 12 cm; angle between tangents ≈ 53.13°

Question 1: A student states: 'A line passing through the center of a circle and touching a point outside the circle is a tangent.' Is this correct? Explain your answer and correct the misconception.

Answer: No, a line passing through the circle's center is a diameter, not a tangent.

Question 2: A common mistake is assuming that the tangent line can intersect the circle at two points. Identify this mistake and explain why it's false.

Answer: Because a tangent touches the circle at exactly one point and does not intersect it at two points.