To solve the problem of finding the [tex]$x$[/tex]-coordinate for point [tex]$E$[/tex], which divides the segment [tex]$\overline{CD}$[/tex] in the ratio [tex]\(3.5\)[/tex], we need to follow these steps:
1. Understand the given ratio: The ratio [tex]\(3.5\)[/tex] can be interpreted more effectively by converting it into a simplest fractional form. [tex]\(3.5\)[/tex] is equivalent to the fraction [tex]\(\frac{7}{2}\)[/tex]. To match this ratio with parts more apt for dividing a segment, we can restate the ratio [tex]\(3.5:1\)[/tex] as [tex]\(3:5\)[/tex].
2. Divide the segment according to the simplified ratio: The simplified ratio [tex]\(3:5\)[/tex] indicates that the segment can be conceptualized as having 3 parts plus 5 parts, making a total of 8 parts.
3. Calculate the fraction related to the distance: To find the distance to point [tex]\(E\)[/tex] from one of the endpoints using this ratio, we must use a fraction of the segment's total distance. In this case, we'll use the fraction related to the part of the distance from point [tex]\(C\)[/tex] to point [tex]\(E\)[/tex].
4. Fraction associated with [tex]\(x\)[/tex]-coordinate: Given the points [tex]\(C (3, 4)\)[/tex] and [tex]\(D (11, 3)\)[/tex] and a distance of 8 units in [tex]\(x\)[/tex]-direction, and a [tex]\(3:5\)[/tex] ratio, we derive:
- The fraction corresponding to the distance from [tex]\(C\)[/tex] to [tex]\(E\)[/tex] is thus:
[tex]\[
\frac{3}{3+5} = \frac{3}{8}
\][/tex]
Therefore, the correct fraction to use to find the [tex]\(x\)[/tex]-coordinate for point [tex]\(E\)[/tex] is [tex]\(\frac{3}{8}\)[/tex].
To summarize the steps clearly:
1. Convert the ratio 3.5 to a simpler ratio [tex]\(3:5\)[/tex].
2. Find the total number of parts as [tex]\(3 + 5 = 8\)[/tex].
3. Use the fraction [tex]\(\frac{3}{8}\)[/tex] to find the portion of the distance for finding [tex]\(E\)[/tex]'s [tex]\(x\)[/tex]-coordinate.
Thus, the correct fraction is:
[tex]\[
\boxed{\frac{3}{8}}
\][/tex]
