Question 1: Calculate the length of the tangent segment from point P to the circle with radius 5 cm if the distance from P to the circle's center is 13 cm.

Answer: 12

Question 2: In a circle with radius 8 meters, a tangent is drawn from point Q outside the circle such that the distance from Q to the circle's center is 10 meters. What is the length of the tangent segment?

Answer: 6

Question 3: A circle has a radius of 9 units. If a point outside the circle is 15 units from the center, what is the length of the tangent from that point to the circle?

Answer: 12

Question 1: A tower stands 20 meters from the edge of a circular pond with radius 15 meters. A cable is stretched from the top of the tower to a point on the pond's edge, forming a tangent to the pond. Calculate the length of the cable, assuming the tower's height is 25 meters.

Answer: Calculate using Pythagoras theorem considering the tangent and the line from tower to pond center.

Question 2: A construction engineer designs a circular arch with radius 10 meters. A support cable is attached from a point 12 meters away from the circle's center, creating a tangent to the circle. Find the length of the tangent cable.

Answer: 8

Question 1: An engineer is designing a wheel with a radius of 1.5 meters. A support arm is attached at a point outside the wheel such that it forms a tangent to the wheel's circumference at a point on the edge. If the support arm is 4 meters from the wheel's center, what is the length of the support arm?

Answer: 3.5

Question 2: In a bridge construction project, a cable runs from a tower 30 meters away from a circular pillar of radius 6 meters, creating a tangent to the pillar. What is the length of the cable from the tower to the tangent point?

Answer: 28

Question 1: A circle has a radius of 7 cm. A point outside the circle is 25 cm from the center. Find the length of the tangent segments from this point to the circle, and verify if the tangent segments are equal.

Answer: 18; Yes, they are equal.

Question 2: Prove that the tangent segments from a common external point to a circle are equal, using geometric properties.

Answer: Explanation involving radii and congruent triangles.

Question 1: Construct a circle with radius 4 cm. From an external point, draw two tangents to the circle. Measure the distance from the external point to the center and verify the tangent lengths.

Answer: Calculate using Pythagoras theorem.

Question 2: Explain why the lengths of tangents from an external point to a circle are always equal.

Answer: Based on properties of congruent triangles formed by radii and tangent lines.

Question 1: A student calculated the tangent length from a point 10 meters from the circle center with radius 6 meters as 4 meters. Identify the error and find the correct length.

Answer: Error: used incorrect formula. Correct length is 8 meters.

Question 2: A student claims that the tangent from an external point can be longer than the distance from the point to the circle's center. Is this possible? Correct the misconception.

Answer: No, tangent length cannot exceed the distance from the external point to the circle's center because of the Pythagorean theorem.