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?

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."