To find the coordinates of point [tex]\( C \)[/tex] that divides the line segment [tex]\( \overline{AB} \)[/tex] in the ratio [tex]\( 1:3 \)[/tex], we use the section formula. The section formula states that if a point [tex]\( C \)[/tex] divides the line segment joining [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]\( C \)[/tex] are given by:
[tex]\[ C\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \][/tex]
Given:
- [tex]\( A \)[/tex] has coordinates [tex]\( (0, 0) \)[/tex]
- [tex]\( B \)[/tex] has coordinates [tex]\( (12, 12) \)[/tex]
- The ratio [tex]\( m:n \)[/tex] is [tex]\( 1:3 \)[/tex]
Substitute the given values into the section formula:
[tex]\[ C_x = \frac{1 \cdot 12 + 3 \cdot 0}{1+3} = \frac{12 + 0}{4} = \frac{12}{4} = 3 \][/tex]
[tex]\[ C_y = \frac{1 \cdot 12 + 3 \cdot 0}{1+3} = \frac{12 + 0}{4} = \frac{12}{4} = 3 \][/tex]
Thus, the coordinates of point [tex]\( C \)[/tex] are [tex]\( (3, 3) \)[/tex].
Comparing with the given options, none of them lists the correct coordinates directly. However, this indicates that the proposed options might not be listing all possible answers.
Therefore, point [tex]\( C \)[/tex] which divides the line segment [tex]\( \overline{AB} \)[/tex] in the ratio [tex]\( 1:3 \)[/tex] has the coordinates [tex]\( (3, 3) \)[/tex].
