To determine the correct equation for the distance between points [tex]\((1,-1)\)[/tex] and [tex]\((-2,2)\)[/tex], let's analyze each of the provided options step-by-step.
### Equation 1
[tex]\[ d = \sqrt{(1-2)^2 + (-1+2)^2} \][/tex]
Subtract the coordinates:
[tex]\[ 1 - 2 = -1 \][/tex]
[tex]\[ -1 + 2 = 1 \][/tex]
Now square these numbers:
[tex]\[ (-1)^2 = 1 \][/tex]
[tex]\[ (1)^2 = 1 \][/tex]
Add the squared terms:
[tex]\[ 1 + 1 = 2 \][/tex]
Take the square root:
[tex]\[ \sqrt{2} \approx 1.4142135623730951 \][/tex]
### Equation 2
[tex]\[ d = \sqrt{(1-2)^2 + (2-1)^2} \][/tex]
Subtract the coordinates:
[tex]\[ 1 - 2 = -1 \][/tex]
[tex]\[ 2 - 1 = 1 \][/tex]
Now square these numbers:
[tex]\[ (-1)^2 = 1 \][/tex]
[tex]\[ (1)^2 = 1 \][/tex]
Add the squared terms:
[tex]\[ 1 + 1 = 2 \][/tex]
Take the square root:
[tex]\[ \sqrt{2} \approx 1.4142135623730951 \][/tex]
### Equation 3
[tex]\[ d = \sqrt{(1+2)^2 + (-1-2)^2} \][/tex]
Add the coordinates:
[tex]\[ 1 + 2 = 3 \][/tex]
[tex]\[ -1 - 2 = -3 \][/tex]
Now square these numbers:
[tex]\[ (3)^2 = 9 \][/tex]
[tex]\[ (-3)^2 = 9 \][/tex]
Add the squared terms:
[tex]\[ 9 + 9 = 18 \][/tex]
Take the square root:
[tex]\[ \sqrt{18} \approx 4.242640687119285 \][/tex]
### Equation 4
[tex]\[ d = \sqrt{(1+2) + (-1-2)} \][/tex]
Add the coordinates:
[tex]\[ 1 + 2 = 3 \][/tex]
[tex]\[ -1 - 2 = -3 \][/tex]
Add these numbers together:
[tex]\[ 3 + (-3) = 0 \][/tex]
Take the square root:
[tex]\[ \sqrt{0} = 0 \][/tex]
Now let's compare the results with the numerical answers from the analysis:
- [tex]\(\sqrt{2} \approx 1.4142135623730951\)[/tex]
- [tex]\(\sqrt{2} \approx 1.4142135623730951\)[/tex]
- [tex]\(\sqrt{18} \approx 4.242640687119285\)[/tex]
- [tex]\(\sqrt{0} = 0\)[/tex]
Given these results, the correct equations that represents the distance between points [tex]\((1,-1)\)[/tex] and [tex]\((-2,2)\)[/tex] are:
[tex]\[ d = \sqrt{(1-2)^2 + (-1+2)^2} \][/tex]
[tex]\[ d = \sqrt{(1-2)^2 + (2-1)^2} \][/tex]
