To determine the ratio in which the point [tex]\((15, 11)\)[/tex] divides the line segment joining the points [tex]\((5, 15)\)[/tex] and [tex]\((20, 9)\)[/tex], we can use the section formula.
The section formula states that a point [tex]\((x, y)\)[/tex] dividing the line segment joining [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] has coordinates:
[tex]\[ \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \][/tex]
Given:
[tex]\[
(x_1, y_1) = (5, 15) \\
(x_2, y_2) = (20, 9) \\
(x, y) = (15, 11)
\][/tex]
We need to find the ratio [tex]\( \frac{m}{n} \)[/tex] such that:
[tex]\[
15 = \frac{m \cdot 20 + n \cdot 5}{m + n} \\
11 = \frac{m \cdot 9 + n \cdot 15}{m + n}
\][/tex]
Let's solve these equations one by one.
1. Solving for [tex]\(x\)[/tex]:
From the [tex]\(x\)[/tex]-coordinate:
[tex]\[
15 = \frac{20m + 5n}{m + n}
\][/tex]
Multiply both sides by [tex]\(m + n\)[/tex]:
[tex]\[
15(m + n) = 20m + 5n \\
15m + 15n = 20m + 5n
\][/tex]
Rearrange to isolate [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[
15n - 5n = 20m - 15m \\
10n = 5m \\
m = 2n
\][/tex]
This implies that the ratio [tex]\( \frac{m}{n} = 2 \)[/tex].
2. Verifying with [tex]\(y\)[/tex]-coordinate:
From the [tex]\(y\)[/tex]-coordinate:
[tex]\[
11 = \frac{9m + 15n}{m + n}
\][/tex]
Again, multiply both sides by [tex]\(m + n\)[/tex]:
[tex]\[
11(m + n) = 9m + 15n \\
11m + 11n = 9m + 15n
\][/tex]
Rearrange to isolate [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[
11m - 9m = 15n - 11n \\
2m = 4n \\
m = 2n
\][/tex]
This confirms that the ratio [tex]\( \frac{m}{n} = 2 \)[/tex].
Thus, 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].
