https://file.aiquickdraw.com/mathbot/file/1726798198912wqo4slaf.jpg

The image you uploaded shows a circle with two intersecting chords and a tangent. The problem asks to find the values of \( x \) and \( y \), which are marked on the chords and tangent. Let me explain how we can solve this problem:

### Step 1: Applying the Tangent-Secant Theorem

The Tangent-Secant Theorem (or Power of a Point Theorem) states that:
\[
EF^2 = DF \cdot FG
\]
Here, \( EF = 9 \), \( DF = 9 \), and \( FG = 3x \).

Substituting these values:
\[
9^2 = 9 \cdot 3x
\]
\[
81 = 27x
\]
\[
x = \frac{81}{27} = 3
\]

### Step 2: Applying the Intersecting Chord Theorem

The Intersecting Chord Theorem states that the products of the segments of two intersecting chords are equal. In this case, for chords \( AB \) and \( CD \):
\[
AH \cdot HB = DH \cdot HG
\]
Given \( AH = 2x \), \( HB = 6 \), \( DH = 9 \), and \( HG = y \), substituting \( x = 3 \) from the previous result:
\[
(2 \cdot 3) \cdot 6 = 9 \cdot y
\]
\[
6 \cdot 6 = 9 \cdot y
\]
\[
36 = 9y
\]
\[
y = \frac{36}{9} = 4
\]

### Final Answer:
- \( x = 3 \)
- \( y = 4 \)

Would you like more details or have any further questions?

Here are five related questions to expand your understanding:

1. How does the Power of a Point Theorem apply to other circle configurations?
2. What happens when two tangents from a point outside a circle are drawn?
3. Can you derive the Intersecting Chord Theorem using geometric properties?
4. What is the relationship between the angle formed by a tangent and a chord at the point of contact?
5. How can you use the Pythagorean Theorem to solve problems involving circles and tangents?

**Tip:** For circle problems, always identify key theorems related to chords, tangents, and secants. This helps simplify seemingly complex problems.

Find the value of x and y based on the intersecting chords and tangent in the circle diagram.

Learn how to find the values of x and y using the Tangent-Secant Theorem and the Intersecting Chord Theorem. The problem involves calculating segment lengths in a circle and applying key geometric principles. Suitable for students in Grades 9-10.

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