To determine the distance between the points (-4, -4) and (3, -1) on a coordinate plane, you can follow these steps:
1. Identify the coordinates of the points:
- Point 1: \((-4, -4)\)
- Point 2: \((3, -1)\)
2. Calculate the differences in the x-coordinates and the y-coordinates:
- Difference in x-coordinates (\(\Delta x\)): \(3 - (-4) = 3 + 4 = 7\)
- Difference in y-coordinates (\(\Delta y\)): \(-1 - (-4) = -1 + 4 = 3\)
3. Apply the Pythagorean Theorem to find the distance:
The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse (in this case, the distance between the two points) is equal to the sum of the squares of the other two sides. Thus, the distance \(d\) can be calculated as:
[tex]\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\][/tex]
Substitute the values of \(\Delta x\) and \(\Delta y\):
[tex]\[
d = \sqrt{(7)^2 + (3)^2}
\][/tex]
[tex]\[
d = \sqrt{49 + 9}
\][/tex]
[tex]\[
d = \sqrt{58}
\][/tex]
4. Calculate the distance:
[tex]\[
d \approx 7.615773105863909
\][/tex]
5. Round the distance to the nearest thousandth:
The nearest thousandth place is the third decimal place. Thus:
[tex]\[
\text{Rounded distance} \approx 7.616
\][/tex]
So, the distance between the points (-4, -4) and (3, -1) is approximately [tex]\(7.616\)[/tex] units when rounded to the nearest thousandth.
