Find the distance [tex] d\left(P_1, P_2\right) [/tex] between the given points [tex] P_1 [/tex] and [tex] P_2 [/tex].

[tex]\[
\begin{aligned}
P_1 &= (-3, 1) \\
P_2 &= (4, 4)
\end{aligned}
\][/tex]

[tex]\[
d\left(P_1, P_2\right) = \boxed{\phantom{d\left(P_1, P_2\right)}}
\][/tex]



Answer :

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).