To find the distance between the points [tex]\((9, -6)\)[/tex] and [tex]\((-4, 7)\)[/tex], we use the distance formula. The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, the coordinates of the first point [tex]\( (x_1, y_1) \)[/tex] are [tex]\( (9, -6) \)[/tex] and the coordinates of the second point [tex]\( (x_2, y_2) \)[/tex] are [tex]\( (-4, 7) \)[/tex].
Step-by-step, we calculate the differences in the coordinates:
[tex]\[
\Delta x = x_2 - x_1 = -4 - 9 = -13
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 7 - (-6) = 7 + 6 = 13
\][/tex]
Next, we substitute [tex]\( \Delta x \)[/tex] and [tex]\( \Delta y \)[/tex] into the distance formula:
[tex]\[
\text{Distance} = \sqrt{(-13)^2 + (13)^2}
\][/tex]
Calculating the squares of [tex]\( \Delta x \)[/tex] and [tex]\( \Delta y \)[/tex]:
[tex]\[
(-13)^2 = 169
\][/tex]
[tex]\[
(13)^2 = 169
\][/tex]
Adding these values together:
[tex]\[
169 + 169 = 338
\][/tex]
Finally, taking the square root of the sum:
[tex]\[
\sqrt{338}
\][/tex]
We can simplify this radical:
[tex]\[
338 = 2 \times 169 = 2 \times 13^2
\][/tex]
[tex]\[
\sqrt{338} = \sqrt{2} \times 13 = 13\sqrt{2}
\][/tex]
Thus, the distance between the points [tex]\( (9, -6) \)[/tex] and [tex]\( (-4, 7) \)[/tex] is:
[tex]\[
13\sqrt{2}
\][/tex]
So the correct answer is:
[tex]\[
\boxed{D) \, 13\sqrt{2}}
\][/tex]
