Answer :
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]
[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]
