To determine the coordinates of point [tex]\(Q\)[/tex] that partitions the segment [tex]\( \overline{AC} \)[/tex] in a [tex]\(4:1\)[/tex] ratio, we will use the section formula for internal division of a line segment.
Given the coordinates of point [tex]\(A\)[/tex] are [tex]\( A(2, -1) \)[/tex] and the coordinates of point [tex]\(C\)[/tex] are [tex]\( C(4, 2) \)[/tex]. The ratio in which point [tex]\(Q\)[/tex] divides the segment is [tex]\( m:n = 4:1 \)[/tex].
The section formula states that if a point [tex]\( Q(x, y) \)[/tex] divides a line segment joining points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( C(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( Q \)[/tex] are given by:
[tex]\[ x = \frac{mx_2 + nx_1}{m + n} \][/tex]
[tex]\[ y = \frac{my_2 + ny_1}{m + n} \][/tex]
Plugging in the values:
- [tex]\( x_1 = 2 \)[/tex], [tex]\( y_1 = -1 \)[/tex] (coordinates of [tex]\( A \)[/tex])
- [tex]\( x_2 = 4 \)[/tex], [tex]\( y_2 = 2 \)[/tex] (coordinates of [tex]\( C \)[/tex])
- [tex]\( m = 4 \)[/tex]
- [tex]\( n = 1 \)[/tex]
First, calculate the x-coordinate of [tex]\( Q \)[/tex]:
[tex]\[
x = \frac{4 \times 4 + 1 \times 2}{4 + 1} = \frac{16 + 2}{5} = \frac{18}{5} = 3.6
\][/tex]
Next, calculate the y-coordinate of [tex]\( Q \)[/tex]:
[tex]\[
y = \frac{4 \times 2 + 1 \times -1}{4 + 1} = \frac{8 - 1}{5} = \frac{7}{5} = 1.4
\][/tex]
Thus, the coordinates of point [tex]\( Q \)[/tex] are:
[tex]\[
Q(3.6, 1.4)
\][/tex]
