The given problem can be solved using the tangent-secant theorem or the Pythagorean Theorem.

Step-by-step solution:

• Let $O$ be the center of the circle, $P$ be the point outside the circle, and $T$ be the point of tangency.
• The radius of the circle is $r = 5$ cm.
• The distance from the center of the circle to point $P$ is $OP = 15$ cm.
• We need to find the length of the tangent segment $PT$.

In a circle, the radius $OT$ is perpendicular to the tangent $PT$, forming a right triangle $OPT$.

Using the Pythagorean theorem in triangle $OPT$: $OP^2 = OT^2 + PT^2$ Substitute the known values: $15^2 = 5^2 + PT^2$ $225 = 25 + PT^2$ $PT^2 = 225 - 25 = 200$ $PT = \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \text{ cm}$

Thus, the length of the tangent segment is $10\sqrt{2}$ cm, or approximately $14.14$ cm.

Would you like further details or have any questions?

Here are some related questions:

1. How does the tangent-secant theorem relate to this problem?
2. What is the relationship between the radius and the tangent line at a point of contact?
3. How would this change if the point $P$ was closer to the circle?
4. What happens if the radius is increased but the distance from $P$ stays the same?
5. Can you derive this using trigonometric identities?

Tip: In geometry problems involving tangents, always remember that the radius drawn to the point of tangency is perpendicular to the tangent.