point P is 15cm away from the center of the circle of radius 5cm. the length of the tangent segment drawn to the circle from point P is 

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.

Point P is 15cm away from the center of the circle of radius 5cm. The length of the tangent segment drawn to the circle from point P is?

Learn how to find the length of the tangent segment from a point 15 cm away from the center of a circle with a 5 cm radius. This problem uses the Pythagorean Theorem and the Tangent-Secant Theorem for a step-by-step solution, suitable for Grades 8-10.

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