Find the coordinates of [tex]\( P \)[/tex] so that [tex]\( P \)[/tex] partitions the segment [tex]\( AB \)[/tex] in the ratio [tex]\( 1: 1 \)[/tex] if [tex]\( A(-4,15) \)[/tex] and [tex]\( B(10,11) \)[/tex].

A. [tex]\((-4,14)\)[/tex]
B. [tex]\((11,-17)\)[/tex]
C. [tex]\((3,13)\)[/tex]
D. [tex]\((7,-2)\)[/tex]



Answer :

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]