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In the figure given below, AB is a chord and XA and XB are tangents of the circle with center O and radius 20. If AB=32cm, and OC = 12 cm. Find length of XB

To find the length of \( XB \), let's analyze the geometry of the situation.

Given:
- \( AB \) is a chord.
- \( XA \) and \( XB \) are tangents from point \( X \).
- The radius \( r = 20 \) cm.
- \( AB = 32 \) cm.
- \( OC = 12 \) cm, where \( C \) is the midpoint of \( AB \).

### Step 1: Find \( AC \) and \( BC \)
Since \( C \) is the midpoint of \( AB \):
\[
AC = BC = \frac{AB}{2} = \frac{32}{2} = 16 \text{ cm}
\]

### Step 2: Apply the Pythagorean Theorem in \( \triangle OCA \)
Since \( O \) is the center of the circle:
\[
OA = \sqrt{OC^2 + AC^2} = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ cm}
\]
So, \( OA \) is indeed the radius.

### Step 3: Use the Tangent-Secant Theorem (or Power of a Point)
The Tangent-Secant Theorem states:
\[
XA^2 = XB^2 = XA \times XB = XC^2 - OC^2
\]
But since \( XA = XB \) (tangents from a point to a circle are equal), let's directly calculate:
\[
XA = \sqrt{OA^2 - OC^2} = \sqrt{20^2 - 12^2} = \sqrt{400 - 144} = \sqrt{256} = 16 \text{ cm}
\]

Thus, \( XB = 16 \) cm.

### Conclusion
The length of \( XB \) is **16 cm**.

Would you like further details or have any questions? Here are five related questions for further exploration:

1. How would the result change if the chord \( AB \) were shorter?
2. What is the relationship between the tangent lengths and the chord's midpoint?
3. Can you derive the power of a point formula used in this context?
4. How would you solve the problem if the circle's radius were unknown?
5. What are the other applications of the Tangent-Secant Theorem in geometry?

**Tip:** Always remember that the tangents from a single external point to a circle are equal in length, which can be very useful in problems involving circles.

Learn how to find the length of XB in a circle geometry problem involving a chord AB = 32 cm, with tangents XA and XB, and OC = 12 cm. Step-by-step solution using the Pythagorean Theorem and Tangent-Secant Theorem.

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