Answer :
To determine the distance between the points [tex]\((-1,-4)\)[/tex] and [tex]\((-9,-8)\)[/tex], we will use the distance formula, which is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are coordinates of the two points.
Given points are:
[tex]\[ (x_1, y_1) = (-1, -4) \quad \text{and} \quad (x_2, y_2) = (-9, -8) \][/tex]
First, calculate the differences in the x and y coordinates:
[tex]\[ \Delta x = x_2 - x_1 = -9 - (-1) = -9 + 1 = -8 \][/tex]
[tex]\[ \Delta y = y_2 - y_1 = -8 - (-4) = -8 + 4 = -4 \][/tex]
Next, calculate the squares of these differences:
[tex]\[ (\Delta x)^2 = (-8)^2 = 64 \][/tex]
[tex]\[ (\Delta y)^2 = (-4)^2 = 16 \][/tex]
Add these squared differences together:
[tex]\[ (\Delta x)^2 + (\Delta y)^2 = 64 + 16 = 80 \][/tex]
Now, take the square root of this sum to find the distance:
[tex]\[ d = \sqrt{80} \][/tex]
We can compare this result to the options provided:
- [tex]\(\sqrt{208} \text{ units}\)[/tex]
- [tex]\(\sqrt{166} \text{ units}\)[/tex]
- [tex]\(\sqrt{124} \text{ units}\)[/tex]
- [tex]\(\sqrt{80} \text{ units}\)[/tex]
The correct answer matches one of the given options:
[tex]\[ \boxed{\sqrt{80} \text{ units}} \][/tex]
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are coordinates of the two points.
Given points are:
[tex]\[ (x_1, y_1) = (-1, -4) \quad \text{and} \quad (x_2, y_2) = (-9, -8) \][/tex]
First, calculate the differences in the x and y coordinates:
[tex]\[ \Delta x = x_2 - x_1 = -9 - (-1) = -9 + 1 = -8 \][/tex]
[tex]\[ \Delta y = y_2 - y_1 = -8 - (-4) = -8 + 4 = -4 \][/tex]
Next, calculate the squares of these differences:
[tex]\[ (\Delta x)^2 = (-8)^2 = 64 \][/tex]
[tex]\[ (\Delta y)^2 = (-4)^2 = 16 \][/tex]
Add these squared differences together:
[tex]\[ (\Delta x)^2 + (\Delta y)^2 = 64 + 16 = 80 \][/tex]
Now, take the square root of this sum to find the distance:
[tex]\[ d = \sqrt{80} \][/tex]
We can compare this result to the options provided:
- [tex]\(\sqrt{208} \text{ units}\)[/tex]
- [tex]\(\sqrt{166} \text{ units}\)[/tex]
- [tex]\(\sqrt{124} \text{ units}\)[/tex]
- [tex]\(\sqrt{80} \text{ units}\)[/tex]
The correct answer matches one of the given options:
[tex]\[ \boxed{\sqrt{80} \text{ units}} \][/tex]
