Find the distance between the points $(a, b)$ and $(-a, -b)$. [d = \sqrt{(-a - a)^2 + (-b - b)^2}.]

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To find the distance between the points $(a, b)$ and $(-a, -b)$, you can use the distance formula, which is given by: $$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Let's assign $(x_1, y_1) = (a, b)$ and $(x_2, y_2) = (-a, -b)$. Plugging these into the distance formula gives:

  1. For the x-coordinates: $$ (-a) - (a) = -2a $$

  2. For the y-coordinates: $$ (-b) - (b) = -2b $$

Then, the distance calculation becomes: $$ \sqrt{(-2a)^2 + (-2b)^2} $$

Simplifying the squares: $$ \sqrt{4a^2 + 4b^2} $$

Factoring out the constant '4' under the square root gives: $$ \sqrt{4(a^2 + b^2)} = 2\sqrt{a^2 + b^2} $$

Thus, the distance between $(a, b)$ and $(-a, -b)$ is: $$ \boxed{2\sqrt{a^2 + b^2}} $$

This formula provides the final answer, establishing the distance between these two points in a Cartesian plane.

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