To calculate the shortest distance between point [tex]\( P(-4, -2) \)[/tex] and point [tex]\( Q(4, 3) \)[/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]
1. Identify the coordinates:
- Coordinates of point [tex]\( P \)[/tex]: [tex]\( (x_1, y_1) = (-4, -2) \)[/tex]
- Coordinates of point [tex]\( Q \)[/tex]: [tex]\( (x_2, y_2) = (4, 3) \)[/tex]
2. Calculate the difference in the x-coordinates and y-coordinates:
[tex]\[
x_2 - x_1 = 4 - (-4) = 4 + 4 = 8
\][/tex]
[tex]\[
y_2 - y_1 = 3 - (-2) = 3 + 2 = 5
\][/tex]
3. Square the differences:
[tex]\[
(x_2 - x_1)^2 = 8^2 = 64
\][/tex]
[tex]\[
(y_2 - y_1)^2 = 5^2 = 25
\][/tex]
4. Sum the squared differences:
[tex]\[
(x_2 - x_1)^2 + (y_2 - y_1)^2 = 64 + 25 = 89
\][/tex]
5. Calculate the square root of the sum to find the distance:
[tex]\[
d = \sqrt{89} \approx 9.433981132056603
\][/tex]
6. Round the distance to 1 decimal place:
[tex]\[
\text{Rounded distance} \approx 9.4
\][/tex]
Therefore, the shortest distance between point [tex]\( P \)[/tex] and point [tex]\( Q \)[/tex] is approximately [tex]\( 9.4 \)[/tex] cm.
