To find the distance between the points [tex]\( A(-2, -4) \)[/tex] and [tex]\( B(7, -1) \)[/tex], we use the distance formula, which is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, [tex]\((x_1, y_1)\)[/tex] are the coordinates of point [tex]\( A \)[/tex] and [tex]\((x_2, y_2)\)[/tex] are the coordinates of point [tex]\( B \)[/tex].
Substitute the coordinates [tex]\( A(-2, -4) \)[/tex] and [tex]\( B(7, -1) \)[/tex] into the formula:
1. Compute the difference in the x-coordinates:
[tex]\[
x_2 - x_1 = 7 - (-2) = 7 + 2 = 9
\][/tex]
2. Compute the difference in the y-coordinates:
[tex]\[
y_2 - y_1 = -1 - (-4) = -1 + 4 = 3
\][/tex]
3. Square these differences:
[tex]\[
(x_2 - x_1)^2 = 9^2 = 81
\][/tex]
[tex]\[
(y_2 - y_1)^2 = 3^2 = 9
\][/tex]
4. Add these squared differences:
[tex]\[
(x_2 - x_1)^2 + (y_2 - y_1)^2 = 81 + 9 = 90
\][/tex]
5. Take the square root of the sum to find the distance:
[tex]\[
d = \sqrt{90}
\][/tex]
The result from calculation shows that the distance [tex]\(d\)[/tex] is approximately:
[tex]\[
d \approx 9.486832980505138
\][/tex]
Therefore, the distance between the points [tex]\( A(-2, -4) \)[/tex] and [tex]\( B(7, -1) \)[/tex] is approximately [tex]\( 9.49 \)[/tex] units.
