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.