To find the distance between the two points [tex]\( P = (3, 1) \)[/tex] and [tex]\( Q = (-3, -7) \)[/tex], we use the distance formula. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a coordinate plane is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, the coordinates are:
- [tex]\( P = (3, 1) \)[/tex] which gives [tex]\( x_1 = 3 \)[/tex] and [tex]\( y_1 = 1 \)[/tex].
- [tex]\( Q = (-3, -7) \)[/tex] which gives [tex]\( x_2 = -3 \)[/tex] and [tex]\( y_2 = -7 \)[/tex].
Substituting these values into the distance formula, we have:
[tex]\[
d = \sqrt{((-3) - 3)^2 + ((-7) - 1)^2}
\][/tex]
Let's simplify the expression inside the square root step by step:
1. Calculate the difference in the x-coordinates:
[tex]\[
x_2 - x_1 = -3 - 3 = -6
\][/tex]
2. Calculate the difference in the y-coordinates:
[tex]\[
y_2 - y_1 = -7 - 1 = -8
\][/tex]
3. Square each of these differences:
[tex]\[
(-6)^2 = 36
\][/tex]
[tex]\[
(-8)^2 = 64
\][/tex]
4. Add the squares of the differences:
[tex]\[
36 + 64 = 100
\][/tex]
5. Finally, take the square root of the sum:
[tex]\[
\sqrt{100} = 10
\][/tex]
Therefore, the distance [tex]\( PQ \)[/tex] is [tex]\( 10.0 \)[/tex].
