Key Concepts and Formulas

• Equilateral Triangle Properties: Centroid, incenter, circumcenter, and orthocenter coincide. $R = 2r$, where $R$ is the circumradius and $r$ is the inradius.
• Distance from a Point to a Line: The distance from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$ is given by $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$.

Step-by-Step Solution

Step 1: Identify the Incenter

• What & Why: The centroid of the equilateral triangle is given to be at the origin (0, 0). Since the centroid and incenter coincide for an equilateral triangle, the incenter is also at (0, 0). This is crucial because the inradius is the perpendicular distance from the incenter to any side.
• Math: Incenter = (0, 0)

Step 2: Calculate the Inradius (r)

• What & Why: The inradius ($r$) is the perpendicular distance from the incenter (0, 0) to the line $x + y = 3$. We need to rewrite the line equation in the standard form to use the distance formula.
• Math: The equation of the line is $x + y - 3 = 0$. So, $A = 1$, $B = 1$, and $C = -3$. Using the distance formula: $r = \frac{|(1)(0) + (1)(0) - 3|}{\sqrt{1^2 + 1^2}}$ $r = \frac{|-3|}{\sqrt{2}}$ $r = \frac{3}{\sqrt{2}}$
• Reasoning: This calculation gives us the value of the inradius, which is essential for finding the circumradius and then the required sum.

Step 3: Calculate the Circumradius (R)

• What & Why: In an equilateral triangle, the circumradius is twice the inradius, i.e., $R = 2r$. We use this relationship to directly find the value of $R$ using the value of $r$ calculated in the previous step.
• Math: $R = 2r = 2 \left(\frac{3}{\sqrt{2}}\right) = \frac{6}{\sqrt{2}}$
• Reasoning: This step leverages the special property of equilateral triangles to relate the two radii, simplifying the calculation.

Step 4: Calculate R + r

• What & Why: The problem asks for the value of $R + r$. We now have the values of both $R$ and $r$, so we simply add them together.
• Math: $R + r = \frac{6}{\sqrt{2}} + \frac{3}{\sqrt{2}} = \frac{9}{\sqrt{2}}$ Rationalizing the denominator: $R + r = \frac{9}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{9\sqrt{2}}{2}$
• Reasoning: This is the final step, combining the values of R and r to obtain the answer.

Common Mistakes & Tips

• Confusing Centers: Remember the coincidence of centroid, incenter, circumcenter, and orthocenter ONLY for equilateral triangles.
• Distance Formula: Ensure the line equation is in the form $Ax + By + C = 0$ before using the distance formula.
• Rationalization: While $\frac{9}{\sqrt{2}}$ is correct, sometimes the options are rationalized, so be comfortable with both forms.

Summary

By recognizing that the centroid is also the incenter, we calculated the inradius as the distance from the origin to the given line. Using the relationship $R = 2r$ for equilateral triangles, we found the circumradius and then calculated $R + r = \frac{9}{\sqrt{2}}$.

Final Answer

The final answer is $\boxed{\frac{9}{\sqrt{2}}}$, which corresponds to option (B).