To find the coordinates of the treasure, we can use the section formula in the coordinate geometry. Given the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], the coordinates that divide the line segment joining these points in the ratio [tex]\(m:n\)[/tex] are given by:
[tex]\[
x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
\][/tex]
### Given data:
- Coordinates of the rock: [tex]\((x_1, y_1) = (5, 7)\)[/tex]
- Coordinates of the tree: [tex]\((x_2, y_2) = (13, 15)\)[/tex]
- Ratio [tex]\(m:n = 5:9\)[/tex]
### Calculation:
1. First, we need to sum the parts of the ratio [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[ m + n = 5 + 9 = 14 \][/tex]
2. Next, we calculate the x-coordinate of the treasure using the section formula:
[tex]\[
x = \left(\frac{5}{14}\right)(13 - 5) + 5
\][/tex]
Simplifying inside the brackets first:
[tex]\[
x = \left(\frac{5}{14}\right) \times 8 + 5
\][/tex]
Calculate the multiplication:
[tex]\[
x = \frac{40}{14} + 5
\][/tex]
Simplify the fraction:
[tex]\[
x = 2.8571 + 5
\][/tex]
Add the values:
[tex]\[
x = 7.8571
\][/tex]
Round to the nearest tenth:
[tex]\[
x \approx 7.9
\][/tex]
3. Now, we calculate the y-coordinate of the treasure:
[tex]\[
y = \left(\frac{5}{14}\right)(15 - 7) + 7
\][/tex]
Simplifying inside the brackets first:
[tex]\[
y = \left(\frac{5}{14}\right) \times 8 + 7
\][/tex]
Calculate the multiplication:
[tex]\[
y = \frac{40}{14} + 7
\][/tex]
Simplify the fraction:
[tex]\[
y = 2.8571 + 7
\][/tex]
Add the values:
[tex]\[
y = 9.8571
\][/tex]
Round to the nearest tenth:
[tex]\[
y \approx 9.9
\][/tex]
### Conclusion:
The coordinates of the treasure, rounded to the nearest tenth, are [tex]\((7.9, 9.9)\)[/tex].
Comparing this to the given options:
[tex]\[
\begin{array}{l}
(11.4, 14.2) \\
(7.6, 8.8) \\
(5.7, 7.5) \\
(10.2, 12.6) \\
\end{array}
\][/tex]
None of the provided options match the correct coordinates of [tex]\((7.9, 9.9)\)[/tex]. Therefore, there might be an error in the provided answer choices.
