To determine the ratio in which the point [tex]\((x, 14)\)[/tex] divides the line segment joining the points [tex]\((7, 11)\)[/tex] and [tex]\((-18, 16)\)[/tex], we can use the section formula. The given points are [tex]\(A(7, 11)\)[/tex] and [tex]\(B(-18, 16)\)[/tex], and the point of division is [tex]\(P(x, 14)\)[/tex] which divides the line segment [tex]\(AB\)[/tex] in the ratio [tex]\(m:n\)[/tex].
### Step 1: Set up the section formula for the y-coordinates
The y-coordinate of the point [tex]\(P\)[/tex] can be expressed using the section formula:
[tex]\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m + n}
\][/tex]
Given [tex]\(y = 14\)[/tex], [tex]\(y_1 = 11\)[/tex], and [tex]\(y_2 = 16\)[/tex], we can set up the equation:
[tex]\[
14 = \frac{m \cdot 16 + n \cdot 11}{m + n}
\][/tex]
### Step 2: Solve for the ratio [tex]\(m:n\)[/tex]
Multiply both sides by [tex]\((m + n)\)[/tex]:
[tex]\[
14(m + n) = 16m + 11n
\][/tex]
Expand and rearrange terms:
[tex]\[
14m + 14n = 16m + 11n
\][/tex]
[tex]\[
14n - 11n = 16m - 14m
\][/tex]
[tex]\[
3n = 2m
\][/tex]
Thus, the ratio [tex]\(m:n = 3:2\)[/tex].
### Step 3: Use the ratio to find the x-coordinate
Now that we know the ratio [tex]\(m:n = 3:2\)[/tex], we use the section formula for the x-coordinate:
[tex]\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m + n}
\][/tex]
Given [tex]\(x_1 = 7\)[/tex] and [tex]\(x_2 = -18\)[/tex], we substitute the ratio [tex]\(m = 3\)[/tex] and [tex]\(n = 2\)[/tex] into the formula:
[tex]\[
x = \frac{3 \cdot (-18) + 2 \cdot 7}{3 + 2}
\][/tex]
Calculate the numerator and denominator:
[tex]\[
x = \frac{-54 + 14}{5}
\][/tex]
[tex]\[
x = \frac{-40}{5}
\][/tex]
[tex]\[
x = -8
\][/tex]
### Conclusion
The point [tex]\((x, 14)\)[/tex] divides the line segment joining the points [tex]\((7, 11)\)[/tex] and [tex]\((-18, 16)\)[/tex] in the ratio [tex]\(3:2\)[/tex]. The x-coordinate of the point is [tex]\(-8\)[/tex].
