The problem asks for the coordinates of point $R$ in the isosceles triangle $PQR$, where:

• $P(0, 0)$
• $Q(2a, 0)$
• $R$ has coordinates $(?, b)$.

We need to determine the $x$-coordinate of $R$, knowing that the triangle is isosceles and symmetric about the vertical axis through $R$.

Step-by-Step Solution:

1. Symmetry Property:
   Since the triangle $PQR$ is isosceles and symmetric about the vertical line passing through $R$, the $x$-coordinate of $R$ lies exactly at the midpoint of $P$ and $Q$.

2. Find Midpoint: The midpoint of $P(0, 0)$ and $Q(2a, 0)$ is calculated as: $x = \frac{x_P + x_Q}{2} = \frac{0 + 2a}{2} = a.$

   Thus, the $x$-coordinate of $R$ is $a$.

3. Verify the Coordinates: The coordinates of $R$ are: $(a, b).$

Final Answer:

The correct choice is B: $(a, b)$.

Do you need further clarification on this solution? Here are some additional questions to expand your understanding:

1. How do you calculate the midpoint of a segment given two points?
2. What is the property of symmetry in isosceles triangles?
3. Can you find the area of triangle $PQR$ with these coordinates?
4. What if $R$ was not symmetric—how would you solve it differently?
5. How would the solution change if the triangle was not isosceles?

Tip: Always look for symmetry when solving geometric problems involving isosceles triangles. It simplifies the calculations significantly.