Line \(XY\) is a secant to the larger circle and is tangent to the smaller circle at point \(Q\). The radius of the smaller circle is \(7\) and the radius of the larger circle is \(9\). What is the length of chord \(XY\)?

\(8\sqrt{2}\approx11.31\)
\(16\)
\(\sqrt{130}\approx11.4\)
\(4\sqrt{2}\approx5.66\)

Given the problem, we can use the Power of a Point theorem to solve for the length of chord \(XY\). 

The Power of a Point theorem states that if a point \(P\) lies outside a circle and two line segments \(PA\) and \(PB\) are drawn from \(P\) to the circle such that \(PA\) and \(PB\) are tangent and secant segments respectively, then the power of the point \(P\) is given by:

\[
PA^2 = PB \cdot PC
\]

In this case, line \(XY\) is a secant to the larger circle and tangent to the smaller circle at point \(Q\). Let's denote:
- \(PQ\) as the tangent from \(P\) to the smaller circle, where \(P\) is the point where the secant touches the larger circle.
- \(O_1\) and \(O_2\) as the centers of the smaller and larger circles, respectively.
- \(r_1 = 7\) as the radius of the smaller circle.
- \(r_2 = 9\) as the radius of the larger circle.
- \(XY = 2a\) as the length of the chord of the larger circle we need to find.

Since \(XY\) is tangent to the smaller circle at \(Q\), \(PQ\) is equal to the radius of the smaller circle, \(7\). By the Power of a Point theorem:

\[
PQ^2 = PX \cdot PY
\]

Let \(PX\) and \(PY\) be the distances from \(P\) to points \(X\) and \(Y\) on the larger circle. Since \(XY\) is the chord, \(PX = PY\), so \(PX \cdot PY = PX^2\). Hence:

\[
PQ^2 = PX^2
\]

Substituting \(PQ = 7\):

\[
7^2 = PX^2
\]

\[
PX = \sqrt{49} = 7
\]

Now, we use the formula for the length of a chord:

\[
XY = 2 \times \sqrt{r_2^2 - PX^2}
\]

Substituting the values:

\[
XY = 2 \times \sqrt{9^2 - 7^2} = 2 \times \sqrt{81 - 49} = 2 \times \sqrt{32} = 2 \times 4\sqrt{2} = 8\sqrt{2}
\]

Thus, the length of chord \(XY\) is \(8\sqrt{2}\), which approximately equals \(11.31\).

The correct answer is:

\[
\boxed{8\sqrt{2}\approx11.31}
\]

Do you want any more details, or have any questions?

Here are 8 related questions:

1. How does the Power of a Point theorem apply to this problem?
2. What is the significance of the tangent and secant in circle geometry?
3. How do you calculate the length of a chord using the radius and distance from the center?
4. What is the general form of the equation used in the Power of a Point theorem?
5. Can the Power of a Point theorem be applied to two tangent circles?
6. How would the solution change if the radius of the larger circle were different?
7. What other theorems in circle geometry are closely related to the Power of a Point?
8. How does this problem illustrate the relationship between secants, tangents, and chords in a circle?

**Tip:** When using the Power of a Point theorem, always remember that the tangent squared equals the product of the secant's external part and the entire secant length.

Learn how to calculate the length of chord XY in circles problem involving secant and tangent lines. Solve using the Power of a Point theorem with detailed step-by-step explanation. Radius of smaller circle is 7, larger circle is 9. Discover circle geometry insights!

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