To determine the coordinates of point [tex]\( P \)[/tex] which partitions the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 1:1 \)[/tex], where [tex]\( A(-4, 15) \)[/tex] and [tex]\( B(10, 11) \)[/tex], we use the section formula for internal division. The section formula states that if a point [tex]\( P(x,y) \)[/tex] divides the 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]\( P \)[/tex] are given by:
[tex]\[
P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
For this problem:
- Point [tex]\( A(x_1, y_1) = (-4, 15) \)[/tex]
- Point [tex]\( B(x_2, y_2) = (10, 11) \)[/tex]
- Ratio [tex]\( m:n = 1:1 \)[/tex]
Substituting these values into the section formula:
[tex]\[
x_P = \frac{m \cdot x_2 + n \cdot x_1}{m+n} = \frac{1 \cdot 10 + 1 \cdot (-4)}{1+1} = \frac{10 - 4}{2} = \frac{6}{2} = 3
\][/tex]
[tex]\[
y_P = \frac{m \cdot y_2 + n \cdot y_1}{m+n} = \frac{1 \cdot 11 + 1 \cdot 15}{1+1} = \frac{11 + 15}{2} = \frac{26}{2} = 13
\][/tex]
Therefore, the coordinates of [tex]\( P \)[/tex] are [tex]\( (3, 13) \)[/tex].
The correct answer is:
C. [tex]\( (3, 13) \)[/tex]
