Question 1: A circular logo has a radius of 5 cm. A tangent line touches the circle at point T. Calculate the length of the tangent segment from an external point P to T, given that P is 8 cm from the circle's center.

Answer: 6.4

Question 2: In a business model, a circular conveyor belt has a radius of 3 meters. A sensor is placed 4 meters from the center on the same line as a tangent point. Calculate the distance from the sensor to the tangent point.

Answer: 3.0

Question 3: Construct a circle with radius 4 cm. Draw a tangent at point T such that a line from the circle's center to T is perpendicular to the tangent. Show all constructions clearly.

Answer: Correct tangent perpendicular to radius at T

Question 4: A circular business advertisement has a radius of 6 meters. A line from an external billboard 10 meters away runs tangent to the circle at point T. Find the length of the tangent segment from the billboard to the point T.

Answer: 8.0

Question 5: Explain why the line from the external point to the point of contact with the circle is perpendicular to the tangent line. Use a business scenario for context.

Answer: Because the tangent line is perpendicular to the radius at the point of contact, ensuring the shortest contact path, similar to how a delivery drone approaches a circular zone at a right angle for efficiency.

Question 6: A circular display stand has a radius of 2 meters. An external point P is located 5 meters from the center. Draw the tangent from P to the circle and calculate the length of the tangent segment.

Answer: 4.0

Question 7: A company plans to hang a circular sign with radius 1.5 meters. The mounting point is 4 meters from the center of the circle. Calculate the length of the tangent from the mounting point to the circle.

Answer: 3.0

Question 8: A business uses a circular track with radius 10 meters for delivery robots. The robots start from a point 12 meters from the center. Draw and measure the tangent from the starting point to the track. Verify the tangent length.

Answer: 8.0

Question 9: Challenge: A circular vault has a radius of 7 meters. An external security camera is located 10 meters from the vault's center. Find the length of the tangent line from the camera to the vault.

Answer: 6.4

Question 10: Error Analysis: A student claims that the tangent from an external point is always longer than the distance from the point to the circle's center. Identify and correct the mistake.

Answer: The student is mistaken; the length of the tangent is always shorter than the distance from the external point to the circle's center, since it forms a right-angled triangle with the radius.