To find the distance between the two points [tex]\((-6, -1)\)[/tex] and [tex]\((-9, -6)\)[/tex], we will follow these steps:
1. Identify the coordinates:
- Point 1: [tex]\((x_1, y_1) = (-6, -1)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (-9, -6)\)[/tex]
2. Calculate the differences between the x and y coordinates:
- [tex]\(\Delta x = x_2 - x_1 = -9 - (-6)\)[/tex]
- [tex]\(\Delta y = y_2 - y_1 = -6 - (-1)\)[/tex]
Performing the calculations:
- [tex]\(\Delta x = -9 + 6 = -3\)[/tex]
- [tex]\(\Delta y = -6 + 1 = -5\)[/tex]
3. Use the Pythagorean theorem to calculate the distance [tex]\(d\)[/tex] between the points:
- The formula for the distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2}
\][/tex]
Substitute [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] into the formula:
- [tex]\[
d = \sqrt{(-3)^2 + (-5)^2}
\][/tex]
Calculate the squares:
- [tex]\[
d = \sqrt{9 + 25}
\][/tex]
Add these values:
- [tex]\[
d = \sqrt{34}
\][/tex]
Calculate the square root:
- [tex]\[
d \approx 5.830951894845301
\][/tex]
4. Round the distance to the nearest tenth:
- The distance [tex]\(d \approx 5.830951894845301\)[/tex] rounded to the nearest tenth is [tex]\(5.8\)[/tex].
Therefore, the distance between the two points [tex]\((-6, -1)\)[/tex] and [tex]\((-9, -6)\)[/tex] is approximately [tex]\(5.8\)[/tex].
