To find the distance between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the Cartesian coordinate system, we use the distance formula. Let's analyze the options provided:
1. Option A: [tex]\(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)[/tex]
This is the correct distance formula. It is derived from the Pythagorean theorem, where the horizontal difference [tex]\((x_2 - x_1)\)[/tex] and the vertical difference [tex]\((y_2 - y_1)\)[/tex] form a right triangle with the distance being the hypotenuse.
2. Option B: [tex]\(\sqrt{(x_2 - x_1)^2 - (y_2 - y_1)^2}\)[/tex]
This formula is incorrect because the subtraction inside the square root can lead to negative values, which aren't valid in measuring distance.
3. Option C: [tex]\(\sqrt{(x_2 + x_1)^2 + (y_2 + y_1)^2}\)[/tex]
This formula does not correctly represent the distance between two points. It combines the coordinates incorrectly.
4. Option D: [tex]\(\sqrt{(x_2 + x_1)^2 - (y_2 - y_1)^2}\)[/tex]
Similar to option B, this formula is incorrect because it can produce negative values inside the square root.
Therefore, the correct formula to circle is:
[tex]\[
\boxed{\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}
\][/tex]
### Part II: Finding the Length of the Segment
Given two points for the endpoints of the segment, [tex]\((1, 2)\)[/tex] and [tex]\((4, 6)\)[/tex]:
1. Identify the coordinates:
- [tex]\( (x_1, y_1) = (1, 2) \)[/tex]
- [tex]\( (x_2, y_2) = (4, 6) \)[/tex]
2. Substitute the coordinates into the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{(4 - 1)^2 + (6 - 2)^2}
\][/tex]
3. Calculate the differences and their squares:
[tex]\[
(4 - 1)^2 = 3^2 = 9
\][/tex]
[tex]\[
(6 - 2)^2 = 4^2 = 16
\][/tex]
4. Add the squares of the differences:
[tex]\[
9 + 16 = 25
\][/tex]
5. Take the square root of the sum:
[tex]\[
\text{Distance} = \sqrt{25} = 5
\][/tex]
Thus, the length of the segment with endpoints [tex]\((1, 2)\)[/tex] and [tex]\((4, 6)\)[/tex] is:
[tex]\[
\boxed{5.0}
\][/tex]
