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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+(2-1)^2}[/tex]
B. [tex]d=\sqrt{(1-2)^2+(-1+2)^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 which equation correctly represents the distance between the points [tex]\((1, -1)\)[/tex] and [tex]\((-2, 2)\)[/tex], we will use the distance formula. The formula for the distance [tex]\( d \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a plane is given by:

[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]

Now, we apply this formula to the given points [tex]\((1, -1)\)[/tex] and [tex]\((-2, 2)\)[/tex]:

1. Calculate the difference in the [tex]\( x \)[/tex]-coordinates:
[tex]\[ x_2 - x_1 = -2 - 1 = -3 \][/tex]

2. Calculate the difference in the [tex]\( y \)[/tex]-coordinates:
[tex]\[ y_2 - y_1 = 2 - (-1) = 2 + 1 = 3 \][/tex]

3. Substitute the values into the distance formula:
[tex]\[ d = \sqrt{(-3)^2 + 3^2} \][/tex]
[tex]\[ d = \sqrt{9 + 9} \][/tex]
[tex]\[ d = \sqrt{18} \][/tex]
[tex]\[ d = 3\sqrt{2} \][/tex]

Now we compare this with each of the given equations:

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

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

3. [tex]\( d = \sqrt{(1+2)^2 + (-1-2)^2} \)[/tex]
[tex]\[ d = \sqrt{(3)^2 + (-3)^2} \][/tex]
[tex]\[ d = \sqrt{9 + 9} \][/tex]
[tex]\[ d = \sqrt{18} \][/tex]
[tex]\[ d = 3\sqrt{2} \][/tex]
This is correct.

4. [tex]\( d = \sqrt{(1+2) + (-1-2)} \)[/tex]
[tex]\[ d = \sqrt{3 + (-3)} \][/tex]
[tex]\[ d = \sqrt{0} \][/tex]
This is incorrect.

Therefore, the correct equation that represents the distance between the points [tex]\((1, -1)\)[/tex] and [tex]\((-2, 2)\)[/tex] is:

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

Thus, the correct answer is:

[tex]\[ \boxed{\sqrt{(1+2)^2+(-1-2)^2}} \][/tex]