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?

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!