To determine in what ratio the point [tex]\( (15, 11) \)[/tex] divides the line segment joining the points [tex]\( (5, 15) \)[/tex] and [tex]\( (20, 9) \)[/tex], we'll employ the section formula. This formula helps find a point that divides a line segment in a specific ratio.
Given the points [tex]\( A = (5, 15) \)[/tex] and [tex]\( B = (20, 9) \)[/tex], and the dividing point [tex]\( P = (15, 11) \)[/tex], let's denote the unknown ratio in which the point divides the segment as [tex]\( m:n \)[/tex].
According to the section formula:
[tex]\[
P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Here, point [tex]\( P = (15, 11) \)[/tex], point [tex]\( A \)[/tex] has coordinates [tex]\( (x_1, y_1) = (5, 15) \)[/tex], and point [tex]\( B \)[/tex] has coordinates [tex]\( (x_2, y_2) = (20, 9) \)[/tex].
Applying the formula, we equate the coordinates of point [tex]\( P \)[/tex] with the expression from the section formula:
[tex]\[
15 = \frac{m \cdot 20 + n \cdot 5}{m + n} \quad \text{(i)}
\][/tex]
[tex]\[
11 = \frac{m \cdot 9 + n \cdot 15}{m + n} \quad \text{(ii)}
\][/tex]
We solve these equations to find the ratio [tex]\( m:n \)[/tex].
1. From equation (i):
[tex]\[
15(m + n) = 20m + 5n
\][/tex]
[tex]\[
15m + 15n = 20m + 5n
\][/tex]
[tex]\[
15n - 5n = 20m - 15m
\][/tex]
[tex]\[
10n = 5m
\][/tex]
[tex]\[
2n = m \quad \text{or} \quad m = 2n \quad \text{(iii)}
\][/tex]
2. Now, from equation (ii):
[tex]\[
11(m + n) = 9m + 15n
\][/tex]
[tex]\[
11m + 11n = 9m + 15n
\][/tex]
[tex]\[
11m - 9m = 15n - 11n
\][/tex]
[tex]\[
2m = 4n
\][/tex]
[tex]\[
m = 2n \quad \text{(iv)}
\][/tex]
From both (iii) and (iv), we find that the ratio [tex]\( m:n \)[/tex] simplifies to:
[tex]\[
m = 2n \quad \Rightarrow \quad \frac{m}{n} = 2 \quad \Rightarrow \quad m:n = 2:1
\][/tex]
So, the point [tex]\( (15, 11) \)[/tex] divides the line segment joining the points [tex]\( (5, 15) \)[/tex] and [tex]\( (20, 9) \)[/tex] in the ratio [tex]\( 2:1 \)[/tex].
