Math / Geometry

QuestionWhat is the length of $\overline{A B}$ ? 6 units 14 units 12 units 10 units

Studdy Solution

STEP 1

1. The length of the line segment $\overline{AB}$ can be calculated using the distance formula in the coordinate plane.
2. The distance formula is derived from the Pythagorean theorem.
3. The coordinates of point $A$ are $(-3, -6)$ , and the coordinates of point $B$ are $(3, 4)$ .

STEP 2

1. Write down the distance formula.
2. Substitute the coordinates of points $A$ and $B$ into the distance formula.
3. Simplify the expression to find the length of $\overline{AB}$ .

STEP 3

Write down the distance formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane.
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

STEP 4

Substitute the coordinates of points $A$ and $B$ into the distance formula. Here, $A = (-3, -6)$ and $B = (3, 4)$ .
$d = \sqrt{(3 - (-3))^2 + (4 - (-6))^2}$

STEP 5

Simplify the differences inside the parentheses.
$d = \sqrt{(3 + 3)^2 + (4 + 6)^2}$

STEP 6

Calculate the sums inside the parentheses.
$d = \sqrt{6^2 + 10^2}$

STEP 7

Square the numbers inside the square root.
$d = \sqrt{36 + 100}$

STEP 8

Add the squared numbers.
$d = \sqrt{136}$

STEP 9

Simplify the square root, if possible. However, $\sqrt{136}$ does not simplify nicely to an integer, but it can be approximated.
$d = \sqrt{136} \approx 11.66$

STEP 10

Compare the approximate length to the provided options. The closest given option is 12 units.
Therefore, the length of $\overline{A B}$ is approximately:
$\boxed{12 \text{ units}}$

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