Question

Which expression represents the distance between point (0, a) and point (a, 0) on a coordinate grid?
StartRoot 2 a squared EndRoot
StartRoot a Superscript 4 Baseline EndRoot
StartRoot 2 a Superscript 4 Baseline EndRoot
0

Answer 1

To find the distance between the points \((0, a)\) and \((a, 0)\) on a coordinate grid, we can use the distance formula, which is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Here, \((x_1, y_1) = (0, a)\) and \((x_2, y_2) = (a, 0)\). Plugging in these coordinates into the distance formula:

\[
d = \sqrt{(a - 0)^2 + (0 - a)^2}
\]

This simplifies to:

\[
d = \sqrt{a^2 + (-a)^2}
\]
\[
d = \sqrt{a^2 + a^2}
\]
\[
d = \sqrt{2a^2}
\]
\[
d = \sqrt{2} \cdot a
\]

None of the options provided directly match this form, which is expressed as \(\sqrt{2} \cdot a\). Instead, let’s look at the given options:

1. \(\sqrt{2a^2}\) simplifies to \(\sqrt{2} \cdot a\)
2. \(\sqrt{a^4}\) simplifies to \(a^2\)
3. \(\sqrt{2a^4}\) simplifies to \(\sqrt{2} \cdot a^2\)
4. \(0\)

Since the expression that best represents the distance derived is \(\sqrt{2} \cdot a\), which can also be written as \(\sqrt{2a^2}\), the correct answer among the options is:

\(\sqrt{2a^2}\)