Question

Which expression can be used to determine the length of segment AB?

On a coordinate plane, triangle A B C has points (4, 3), (negative 2, 1), (negative 2, 3).
StartRoot 2 squared + 6 squared EndRoot
StartRoot 2 squared minus 6 squared EndRoot
StartRoot 6 squared minus 2 squared EndRoot
StartRoot (2 + 6) squared EndRoot

Answer 1

To find the length of segment AB in triangle ABC with points A(4, 3) and B(-2, 1), we can use the distance formula:

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

Where \((x_1, y_1)\) are the coordinates of point A, and \((x_2, y_2)\) are the coordinates of point B.

Here:
- \(x_1 = 4\), \(y_1 = 3\)
- \(x_2 = -2\), \(y_2 = 1\)

Substituting the values into the distance formula:

\[
AB = \sqrt{((-2) - 4)^2 + (1 - 3)^2}
\]

Calculating:

\[
AB = \sqrt{(-6)^2 + (-2)^2}
\]
\[
AB = \sqrt{36 + 4}
\]
\[
AB = \sqrt{40}
\]

Thus, \(AB = \sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}\).

Based on the initial expressions listed in your question, none of them match exactly with the calculation shown. However, we can check if any of the expressions can be simplified or correspond correctly:

- **StartRoot 2 squared + 6 squared EndRoot** simplifies to \(\sqrt{4 + 36} = \sqrt{40}\), which is equivalent to \(AB\).
- **StartRoot 2 squared minus 6 squared EndRoot** simplifies to \(\sqrt{4 - 36} = \sqrt{-32}\), which doesn't apply.
- **StartRoot 6 squared minus 2 squared EndRoot** simplifies to \(\sqrt{36 - 4} = \sqrt{32}\), which doesn't match.
- **StartRoot (2 + 6) squared EndRoot** simplifies to \(\sqrt{8^2} = 8\), which also doesn't match.

Thus, the correct expression to determine the length of segment AB from the options you gave is:

**StartRoot 2 squared + 6 squared EndRoot**