https://file.aiquickdraw.com/mathbot/file/17274279355678gwskjmi.jpg

The problem in the image is asking for the radius of the circumcircle (the circle that passes through all the vertices) of a triangle \( \triangle ABC \), where the following are given:

- \( BC = 3\sqrt{2} \text{ cm} \)
- \( \angle BAC = 45^\circ \)

We are asked to find the radius of the circumcircle. Let's solve it step by step.

### Step 1: Formula for Circumradius
For any triangle, the radius \( R \) of the circumcircle can be found using the following formula:
\[
R = \frac{a}{2 \sin A}
\]
where:
- \( a \) is the side opposite angle \( A \) (in this case, \( BC \))
- \( A \) is the angle opposite side \( a \) (in this case, \( \angle BAC \))

### Step 2: Apply the Given Information
We are given:
- \( a = BC = 3\sqrt{2} \, \text{cm} \)
- \( \angle A = \angle BAC = 45^\circ \)

### Step 3: Calculate the Radius
We need to substitute these values into the formula. Since \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), we can calculate the circumradius \( R \).

\[
R = \frac{3\sqrt{2}}{2 \cdot \frac{\sqrt{2}}{2}} = \frac{3\sqrt{2}}{\sqrt{2}} = 3 \, \text{cm}
\]

Thus, the radius of the circumcircle is **3 cm**.

Would you like a more detailed explanation, or do you have any other questions?

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Here are 5 related questions to expand on the topic:

1. What is the formula for the circumradius if only the sides of a triangle are known?
2. How would you calculate the circumradius for an equilateral triangle?
3. What is the difference between the circumradius and the inradius of a triangle?
4. How is the circumcenter of a triangle geometrically constructed?
5. How does the formula for circumradius change for obtuse triangles?

**Tip:** The circumradius formula \( R = \frac{a}{2 \sin A} \) can be derived using the Law of Sines!

For triangle ABC, it is given that BC = 3√2 cm and angle BAC = 45°. What is the radius of the circle described around triangle ABC?

Discover how to calculate the circumradius of a triangle when one angle is 45° and the opposite side measures 3√2 cm. This example utilizes the formula for the circumradius which is based on the sine of the angle, demonstrating a practical application of trigonometry and geometry.

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