To find the coordinates of point [tex]\(T\)[/tex] that divides the line segment [tex]\(\overline{ RS }\)[/tex] into a [tex]\(4:5\)[/tex] ratio, we use the section formula. The section formula states that if a point [tex]\(T(x, y)\)[/tex] divides a line segment joining points [tex]\(A(x_1, y_1)\)[/tex] and [tex]\(B(x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex], then the coordinates of [tex]\(T\)[/tex] are given by:
[tex]\[
T \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Here, the endpoints [tex]\(R\)[/tex] and [tex]\(S\)[/tex] are given as:
- [tex]\(R(-5, 12)\)[/tex]
- [tex]\(S(4, -6)\)[/tex]
The ratio in which [tex]\(T\)[/tex] divides the segment is [tex]\(4:5\)[/tex], thus [tex]\(m = 4\)[/tex] and [tex]\(n = 5\)[/tex].
We now apply the section formula to find the coordinates of [tex]\(T\)[/tex].
First, we calculate the x-coordinate of [tex]\(T\)[/tex]:
[tex]\[
T_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} = \frac{4 \cdot 4 + 5 \cdot (-5)}{4 + 5} = \frac{16 - 25}{9} = \frac{-9}{9} = -1
\][/tex]
Next, we calculate the y-coordinate of [tex]\(T\)[/tex]:
[tex]\[
T_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} = \frac{4 \cdot (-6) + 5 \cdot 12}{4 + 5} = \frac{-24 + 60}{9} = \frac{36}{9} = 4
\][/tex]
Therefore, the coordinates of point [tex]\(T\)[/tex] are:
[tex]\[
(-1, 4)
\][/tex]
So, the correct answer is [tex]\(\boxed{(-1, 4)}\)[/tex].
