To find the distance [tex]\( d \)[/tex] between the points [tex]\( P_1 = (-3, 1) \)[/tex] and [tex]\( P_2 = (4, 4) \)[/tex], we can use the distance formula for two points in a Cartesian plane. The distance formula is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here's the step-by-step solution:
1. Identify the coordinates of the points:
[tex]\[
P_1 = (x_1, y_1) = (-3, 1)
\][/tex]
[tex]\[
P_2 = (x_2, y_2) = (4, 4)
\][/tex]
2. Calculate the differences in the x-coordinates and y-coordinates:
[tex]\[
\Delta x = x_2 - x_1 = 4 - (-3) = 4 + 3 = 7
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 4 - 1 = 3
\][/tex]
3. Substitute [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] into the distance formula:
[tex]\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{7^2 + 3^2}
\][/tex]
4. Compute the squares of [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex]:
[tex]\[
7^2 = 49
\][/tex]
[tex]\[
3^2 = 9
\][/tex]
5. Add the squares:
[tex]\[
49 + 9 = 58
\][/tex]
6. Take the square root of the sum:
[tex]\[
d = \sqrt{58} \approx 7.615773105863909
\][/tex]
Thus, the distance [tex]\( d \)[/tex] between the points [tex]\( P_1 \)[/tex] and [tex]\( P_2 \)[/tex] is:
[tex]\[
d \approx 7.62
\][/tex] (rounded to two decimal places).
