To solve this problem, we need to find the coordinates of point [tex]\(E\)[/tex] given that point [tex]\(F\)[/tex] lies on segment [tex]\(D E\)[/tex] and the ratio [tex]\(DF: FE\)[/tex] is [tex]\(4:2\)[/tex].
### Step-by-Step Solution:
1. Identify the coordinates of points [tex]\(D\)[/tex] and [tex]\(F\)[/tex]:
- Point [tex]\(D = (-2, -5)\)[/tex]
- Point [tex]\(F = (5, 3)\)[/tex]
2. Understand the ratio:
- The ratio [tex]\(DF: FE = 4:2\)[/tex] can be simplified to [tex]\(2:1\)[/tex]. This means that segment [tex]\(FE\)[/tex] is half the length of segment [tex]\(DF\)[/tex].
3. Total parts in the ratio:
- Since [tex]\(DF\)[/tex] and [tex]\(FE\)[/tex] are in the ratio [tex]\(4:2\)[/tex], there are a total of [tex]\(4 + 2 = 6\)[/tex] parts.
4. Calculate the length of one part in terms of coordinates:
- Each part is [tex]\(1/6\)[/tex] of the entire segment [tex]\(DE\)[/tex].
5. Set up the linear combination for point [tex]\(E\)[/tex]:
- The coordinates of point [tex]\(E\)[/tex] can be determined by extending from point [tex]\(F\)[/tex] in the direction away from [tex]\(D\)[/tex] as the ratio [tex]\(4:2\)[/tex] (or equivalently [tex]\(2:1\)[/tex]) prescribes extending twice the length of [tex]\(FE\)[/tex] that we travel from [tex]\(F\)[/tex] to [tex]\(D\)[/tex].
6. Calculate the x-coordinate of [tex]\(E\)[/tex]:
- The change in x from [tex]\(D\)[/tex] to [tex]\(F\)[/tex] is [tex]\(5 - (-2) = 7\)[/tex].
- Therefore, [tex]\(E_x\)[/tex] will extend [tex]\(2/4\)[/tex] of this difference past [tex]\(F\)[/tex], giving
[tex]\[
E_x = 5 + \left(\frac{1}{2} \cdot 7\right) = 5 + 3.5 = 8.5
\][/tex]
7. Calculate the y-coordinate of [tex]\(E\)[/tex]:
- The change in y from [tex]\(D\)[/tex] to [tex]\(F\)[/tex] is [tex]\(3 - (-5) = 8\)[/tex].
- Therefore, [tex]\(E_y\)[/tex] will extend [tex]\(2/4\)[/tex] of this difference past [tex]\(F\)[/tex], giving
[tex]\[
E_y = 3 + \left(\frac{1}{2} \cdot 8\right) = 3 + 4 = 7
\][/tex]
8. Combine the coordinates:
- Therefore, the coordinates of point [tex]\(E\)[/tex] are [tex]\((8.5, 7)\)[/tex].
### Conclusion:
Thus, the coordinates of point [tex]\(E\)[/tex] are [tex]\((8.5, 7)\)[/tex].
The correct answer is:
[tex]\[
(8.5, 7)
\][/tex]
