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12. Which of the following equations correctly represents the distance between the points [tex]\((1, -1)\)[/tex] and [tex]\((-2, 2)\)[/tex]?

A. [tex]\(d = \sqrt{(1 - 2)^2 + (-1 + 2)^2}\)[/tex]

B. [tex]\(d = \sqrt{(1 - 2)^2 + (2 - 1)^2}\)[/tex]

C. [tex]\(d = \sqrt{(1 + 2)^2 + (-1 - 2)^2}\)[/tex]

D. [tex]\(d = \sqrt{(1 + 2) + (-1 - 2)}\)[/tex]

Sagot :

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]