To find the distance between points [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] with coordinates [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 Cartesian plane is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
For the given points [tex]\( P = (3, 1) \)[/tex] and [tex]\( Q = (-3, -7) \)[/tex]:
- [tex]\( x_1 = 3 \)[/tex]
- [tex]\( y_1 = 1 \)[/tex]
- [tex]\( x_2 = -3 \)[/tex]
- [tex]\( y_2 = -7 \)[/tex]
Substitute these values into the distance formula:
[tex]\[
d = \sqrt{((-3) - 3)^2 + ((-7) - 1)^2}
\][/tex]
First, calculate the differences [tex]\( x_2 - x_1 \)[/tex] and [tex]\( y_2 - y_1 \)[/tex]:
[tex]\[
x_2 - x_1 = -3 - 3 = -6
\][/tex]
[tex]\[
y_2 - y_1 = -7 - 1 = -8
\][/tex]
Next, square these differences:
[tex]\[
(-6)^2 = 36
\][/tex]
[tex]\[
(-8)^2 = 64
\][/tex]
Now, add the squared differences:
[tex]\[
36 + 64 = 100
\][/tex]
Finally, take the square root of the sum:
[tex]\[
d = \sqrt{100} = 10
\][/tex]
Therefore, the distance [tex]\( PQ \)[/tex] is:
[tex]\[
\boxed{10}
\][/tex]
