Absolutely, let's walk through the process of calculating the distance between the points [tex]\((-1, 0)\)[/tex] and [tex]\((-9, 6)\)[/tex] step by step.
### Step 1: Identify the coordinates of the points
We have two points in a Cartesian plane:
- Point 1 [tex]\((x_1, y_1) = (-1, 0)\)[/tex]
- Point 2 [tex]\((x_2, y_2) = (-9, 6)\)[/tex]
### Step 2: Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] coordinates
To find the differences in coordinates, we subtract the corresponding coordinates of the two points:
- Difference in [tex]\(x\)[/tex] coordinates ([tex]\(\Delta x\)[/tex]) = [tex]\(x_2 - x_1\)[/tex]
[tex]\[ \Delta x = -9 - (-1) = -9 + 1 = -8 \][/tex]
- Difference in [tex]\(y\)[/tex] coordinates ([tex]\(\Delta y\)[/tex]) = [tex]\(y_2 - y_1\)[/tex]
[tex]\[ \Delta y = 6 - 0 = 6 \][/tex]
### Step 3: Apply 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] is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the differences we calculated:
[tex]\[ d = \sqrt{(-8)^2 + 6^2} \][/tex]
### Step 4: Compute the squares of the differences
- [tex]\((-8)^2 = 64\)[/tex]
- [tex]\(6^2 = 36\)[/tex]
### Step 5: Sum the squares
[tex]\[ 64 + 36 = 100 \][/tex]
### Step 6: Take the square root of the sum
[tex]\[ d = \sqrt{100} = 10.0 \][/tex]
### Final Answer
The distance between the points [tex]\((-1, 0)\)[/tex] and [tex]\((-9, 6)\)[/tex] is [tex]\(10.0\)[/tex] units.
