Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
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]
[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]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.