To find the distance between the points [tex]\((\frac{5}{2}, 3)\)[/tex] and [tex]\((\frac{3}{2}, 8)\)[/tex], follow these steps:
1. Identify the coordinates of the points:
- Point 1: [tex]\((x_1, y_1) = (\frac{5}{2}, 3)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (\frac{3}{2}, 8)\)[/tex]
2. Calculate the difference in the x-coordinates (Δx) and the y-coordinates (Δy):
[tex]\[
\Delta x = x_2 - x_1 = \frac{3}{2} - \frac{5}{2} = -1
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 8 - 3 = 5
\][/tex]
3. Square these differences:
[tex]\[
(\Delta x)^2 = (-1)^2 = 1
\][/tex]
[tex]\[
(\Delta y)^2 = 5^2 = 25
\][/tex]
4. Sum these squared differences to find the squared distance:
[tex]\[
(\Delta x)^2 + (\Delta y)^2 = 1 + 25 = 26
\][/tex]
5. Take the square root of this sum to find the Euclidean distance between the points:
[tex]\[
\text{Distance} = \sqrt{26}
\][/tex]
Thus, the distance between the points [tex]\((\frac{5}{2}, 3)\)[/tex] and [tex]\((\frac{3}{2}, 8)\)[/tex] is [tex]\(\sqrt{26}\)[/tex].
Therefore, the correct answer is:
A) [tex]\(\sqrt{26}\)[/tex]
