https://file.aiquickdraw.com/mathbot/file/1730625001895tj7rotnm.jpg

Let's go through each part of the problem step-by-step based on the image provided.

### Problem Analysis
You have a circle with center \( A \) and two chords intersecting at \( A \), forming segments with the following lengths:
- \( MA = 2x \)
- \( GA = 3 \)
- \( AT = 28 \)
- \( AY = 7 \)

The questions ask:
1. **2.1** - To identify the pair of segments for each chord.
2. **2.2** - To find the length of segments \( MA \), \( AP \), \( GA \), and \( AY \).
3. **2.3** - To solve for \( x \) using the relation \( MA \cdot AT = GA \cdot AY \).

Let's solve each part.

### Step-by-Step Solution

#### Part 2.1: Identifying the Pairs of Segments for Each Chord
In this case:
- The chord \( MT \) has segments \( MA \) and \( AT \).
- The chord \( GY \) has segments \( GA \) and \( AY \).

#### Part 2.2: Lengths of the Segments
Given:
- \( MA = 2x \)
- \( GA = 3 \)
- \( AT = 28 \)
- \( AY = 7 \)

The lengths are as stated, but we will find \( x \) in the next step to determine the actual length of \( MA \).

#### Part 2.3: Solving for \( x \) Using the Relation
The relation given is:
\[
MA \cdot AT = GA \cdot AY
\]
Substitute the values:
\[
(2x) \cdot 28 = 3 \cdot 7
\]
This simplifies to:
\[
56x = 21
\]
Now, solve for \( x \):
\[
x = \frac{21}{56} = 0.375
\]

### Final Answer
1. \( x = 0.375 \)
2. Lengths of the segments:
   - \( MA = 2x = 2 \cdot 0.375 = 0.75 \)
   - \( GA = 3 \)
   - \( AT = 28 \)
   - \( AY = 7 \)

Would you like more details or have any questions?

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### Additional Questions for Practice
1. If \( x = 0.375 \), what would be the length of the entire chord \( MT \)?
2. How would the calculation change if \( GA \) were doubled?
3. What is the ratio of the segments \( MA \) and \( AT \)?
4. What type of geometric theorem applies in part 2.3?
5. If \( AY \) were 10 instead of 7, how would that affect the value of \( x \)?

### Tip
For intersecting chords within a circle, remember that the product of the segments of one chord equals the product of the segments of the other chord. This is often referred to as the "Intersecting Chord Theorem."

In circle A, tell the pair of segments of each chord. What is the length of segments MA, AP, GA, and AY? Solve for X in the relation MA * AT = GA * AY.

Learn how to solve for X in a geometry problem involving intersecting chords in a circle. This detailed solution explains how to identify segment pairs, calculate segment lengths, and apply the Intersecting Chord Theorem to find the unknown variable X. Suitable for students in Grades 8-10.

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