https://r2.erweima.ai/mathbot/image/17256695358280etf6bfu.jpg

I have opened the image you provided. Let's now analyze and solve the mathematical problems depicted in it.

### First Problem:
The question involves a circle and asks about the length of an arc of a circle given its central angle and radius:
1. The angle \(\theta = 45^\circ\).
2. The radius \(r = 12 \, \text{cm}\).

To calculate the length of the arc (\(s\)), we can use the formula:
\[
s = \frac{\theta}{360^\circ} \times 2\pi r
\]

Let's compute this.

### Second Problem:
In the second diagram, we have a circle with chords and segments. It mentions:
1. \( AB \perp CD \) (AB is perpendicular to CD).
2. \( AB = 15 \, \text{cm}\), \( CD = 24 \, \text{cm}\), and \( OB = 17 \, \text{cm}\).

The question is asking for the measure of the radius of the circle. We can use Pythagoras’ theorem or geometric properties of chords to solve this.

I'll now compute the solution for both problems.### First Problem Solution:
The length of the arc is approximately \( 9.42 \, \text{cm} \).

Now let's move on to solving the second problem, where we need to find the radius of the circle. The given data suggests using geometric properties of chords and right triangles. I'll calculate that next.### Second Problem Solution:
The radius of the circle is approximately \( 22.67 \, \text{cm} \).

If you have any questions or would like further details on how these solutions were derived, feel free to ask!

Here are 5 related questions to expand your understanding:
1. How do we derive the formula for the length of an arc in a circle?
2. What are the properties of chords in a circle?
3. Can you calculate the area of the sector formed by a \(45^\circ\) angle?
4. How would the arc length change if the radius were doubled?
5. How does the Pythagorean theorem apply in circle geometry?

**Tip:** When working with circles, always keep in mind the relationship between angles and arcs, as well as the properties of right triangles in geometric figures.

The query submitted by the user involves two mathematical problems related to circles. The first problem asks about the length of an arc given a central angle and radius, and the second problem involves finding the radius of a circle given certain chord lengths and distances.

Learn how to calculate the length of an arc and the radius of a circle using the arc length formula and the Pythagorean theorem. This geometry problem involves key concepts like circles, arcs, and chords, suitable for students in Grades 9-11.

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