To determine the shortest distance between point P, which has coordinates [tex]\((-4, -2)\)[/tex], and point Q, which has coordinates [tex]\((4, 3)\)[/tex], we use the Euclidean distance formula. The Euclidean distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a plane is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here are the coordinates for points P and Q:
- [tex]\( P: (x_1, y_1) = (-4, -2) \)[/tex]
- [tex]\( Q: (x_2, y_2) = (4, 3) \)[/tex]
Step-by-step calculation:
1. Calculate the difference in the x-coordinates:
[tex]\[ x_2 - x_1 = 4 - (-4) = 4 + 4 = 8 \][/tex]
2. Calculate the difference in the y-coordinates:
[tex]\[ y_2 - y_1 = 3 - (-2) = 3 + 2 = 5 \][/tex]
3. Square both differences:
[tex]\[ (x_2 - x_1)^2 = 8^2 = 64 \][/tex]
[tex]\[ (y_2 - y_1)^2 = 5^2 = 25 \][/tex]
4. Add the squared differences:
[tex]\[ (x_2 - x_1)^2 + (y_2 - y_1)^2 = 64 + 25 = 89 \][/tex]
5. Take the square root of the sum to find the Euclidean distance:
[tex]\[ d = \sqrt{89} \][/tex]
6. Round the result to one decimal place:
Using a calculator, [tex]\(\sqrt{89} \approx 9.433981132\)[/tex]. Rounded to one decimal place, [tex]\(d \approx 9.4\)[/tex].
Therefore, the shortest distance between points P [tex]\((-4, -2)\)[/tex] and Q [tex]\((4, 3)\)[/tex] is approximately 9.4.
