To find the coordinates of point [tex]\( T \)[/tex] that partitions the segment [tex]\(\overline{DF}\)[/tex] with endpoints [tex]\(D(1, 4)\)[/tex] and [tex]\(F(7, 1)\)[/tex] in a [tex]\(3:1\)[/tex] ratio, we use the section formula. This formula helps us determine the coordinates of a point dividing a line segment internally in the given ratio.
The section formula for a point [tex]\( T(x, y) \)[/tex] dividing the segment joining [tex]\(D(x_1, y_1)\)[/tex] and [tex]\(F(x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
T_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \quad T_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
Here, [tex]\( (x_1, y_1) = (1, 4) \)[/tex] are the coordinates of [tex]\( D \)[/tex], [tex]\( (x_2, y_2) = (7, 1) \)[/tex] are the coordinates of [tex]\( F \)[/tex], and the ratio [tex]\( m:n \)[/tex] is [tex]\( 3:1 \)[/tex].
Plugging in the given values:
[tex]\[
T_x = \frac{3 \cdot 7 + 1 \cdot 1}{3 + 1} = \frac{21 + 1}{4} = \frac{22}{4} = 5.5
\][/tex]
[tex]\[
T_y = \frac{3 \cdot 1 + 1 \cdot 4}{3 + 1} = \frac{3 + 4}{4} = \frac{7}{4} = 1.75
\][/tex]
Therefore, the coordinates of point [tex]\( T \)[/tex] are:
[tex]\[
T(5.5, 1.75)
\][/tex]
Thus, point [tex]\( T \)[/tex] located at [tex]\( (5.5, 1.75) \)[/tex] divides the line segment [tex]\(\overline{DF}\)[/tex] in the ratio of [tex]\( 3:1 \)[/tex].
