To calculate the distance between the points [tex]\( K = (0, -9) \)[/tex] and [tex]\( M = (7, -2) \)[/tex] in the coordinate plane, we use the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, our points are [tex]\( K = (0, -9) \)[/tex] and [tex]\( M = (7, -2) \)[/tex].
1. First, we calculate the difference in the x-coordinates:
[tex]\[
x_2 - x_1 = 7 - 0 = 7
\][/tex]
2. Next, we calculate the difference in the y-coordinates:
[tex]\[
y_2 - y_1 = -2 - (-9) = -2 + 9 = 7
\][/tex]
3. Then, we substitute these differences into the distance formula:
[tex]\[
d = \sqrt{(7)^2 + (7)^2}
\][/tex]
4. We simplify the expression inside the square root:
[tex]\[
d = \sqrt{49 + 49}
\][/tex]
5. This simplifies further to:
[tex]\[
d = \sqrt{98}
\][/tex]
6. We can factorize [tex]\( 98 \)[/tex]:
[tex]\[
d = \sqrt{49 \times 2}
\][/tex]
7. Recognizing [tex]\( \sqrt{49} \)[/tex] as [tex]\( 7 \)[/tex], we get:
[tex]\[
d = 7\sqrt{2}
\][/tex]
Therefore, the exact distance between the points [tex]\( K = (0, -9) \)[/tex] and [tex]\( M = (7, -2) \)[/tex] is:
[tex]\[
\boxed{7\sqrt{2}}
\][/tex]
