To determine the ratios in which the points [tex]\(E(1, 2)\)[/tex], [tex]\(D(2, 4)\)[/tex], and [tex]\(C(-3, -6)\)[/tex] divide the line segment [tex]\(\overline{AB}\)[/tex] with endpoints [tex]\(A(-7, -14)\)[/tex] and [tex]\(B(5, 10)\)[/tex], we need to match each point with its respective dividing ratio. Based on given coordinates, the correct pairs are as follows:
- The point [tex]\(E(1, 2)\)[/tex] divides the segment [tex]\(\overline{AB}\)[/tex] in the ratio 1:1.
- The point [tex]\(D(2, 4)\)[/tex] divides the segment [tex]\(\overline{AB}\)[/tex] in the ratio 1:1.
- The point [tex]\(C(-3, -6)\)[/tex] divides the segment [tex]\(\overline{AB}\)[/tex] in the ratio 1:3.
Therefore, the correct pairs would be:
- [tex]$E(1, 2)$[/tex] ⟵⟶ [tex]$1:1$[/tex]
- [tex]$D(2, 4)$[/tex] ⟵⟶ [tex]$1:1$[/tex]
- [tex]$C(-3, -6)$[/tex] ⟵⟶ [tex]$1:3$[/tex]
The points [tex]\(E\)[/tex], [tex]\(D\)[/tex], and [tex]\(C\)[/tex] thus divide the line segment [tex]\(\overline{AB}\)[/tex] into the specified ratios respectively.
