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The problem gives us a circle \( U \), with \( UV = 7 \) and \( m \angle VUW = 45^\circ \), asking for the length of the arc \( VW \) as a fraction times \( \pi \).

### Step-by-step solution:

1. **Radius of the Circle**: 
   Since \( UV = 7 \), this is the radius of the circle. Therefore, the radius \( r \) of the circle is \( 7 \).

2. **Central Angle and Arc Length Formula**:
   The angle subtended by the arc \( VW \) at the center \( U \) is \( 45^\circ \).
   The formula for the length of an arc is:
   \[
   \text{Arc length} = r \times \theta
   \]
   where \( \theta \) is in radians. To convert \( 45^\circ \) to radians:
   \[
   45^\circ = \frac{45 \times \pi}{180} = \frac{\pi}{4} \, \text{radians}
   \]

3. **Calculating the Arc Length**:
   Now, using the formula for the arc length:
   \[
   \text{Arc length of } VW = 7 \times \frac{\pi}{4} = \frac{7\pi}{4}
   \]

Thus, the length of the arc \( VW \) is \( \frac{7\pi}{4} \).

Would you like more details or have any further questions?

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Here are 5 related questions you could explore:
1. How does the central angle affect the length of the arc?
2. What is the total circumference of the circle in this problem?
3. How do you convert degrees to radians in general?
4. How would the arc length change if the angle were doubled?
5. How do arc length formulas differ for different types of curves?

**Tip**: When dealing with angles in degrees and arc lengths, always remember to convert the angle to radians before using the arc length formula!

In circle U, UV = 7 and m∠VUW = 45°. Find the length of arc VW. Express your answer as a fraction times π.

Learn how to calculate the arc length in a circle with radius 7 and a 45-degree central angle. This geometry problem explores the relationship between arc length, radius, and radians, using the formula Arc length = r × θ. Suitable for students in Grades 9-10.

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