A treasure map indicates that a treasure is buried so that it partitions the distance between a rock and a tree in a 5:9 ratio. Marina traced the map onto a coordinate plane to find the exact location of the treasure.

Given formulas:
[tex]\[
\begin{array}{l}
x = \left(\frac{m}{m+n}\right)\left(x_2-x_1\right) + x_1 \\
y = \left(\frac{m}{m+n}\right)\left(y_2-y_1\right) + y_1
\end{array}
\][/tex]

What are the coordinates of the treasure? Round to the nearest tenth if necessary.

A. (11.4, 14.2)
B. (7.6, 8.8)
C. (5.7, 7.5)
D. (10.2, 12.6)



Answer :

Let's determine the coordinates of the treasure using the provided partition ratios and coordinates.

1. First, let's set up the information we have:
- The coordinates of the rock are [tex]\((x_1, y_1) = (0, 0)\)[/tex].
- The coordinates of the tree are [tex]\((x_2, y_2) = (10, 15)\)[/tex].
- The ratio of the distances is [tex]\(m:n = 5:9\)[/tex].

2. The formula to find the coordinates that partition the segment in the given ratio is:
[tex]\[ \left( \frac{m}{m+n}(x_2 - x_1) + x_1, \frac{m}{m+n}(y_2 - y_1) + y_1 \right) \][/tex]

3. Plug the values [tex]\(m = 5\)[/tex] and [tex]\(n = 9\)[/tex] into the formula:

Coordinates:

[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]

4. Calculate the x-coordinate of the treasure:

[tex]\[ x = \left( \frac{5}{5+9} \right) (10 - 0) + 0 = \left( \frac{5}{14} \right) \cdot 10 \][/tex]
[tex]\[ x = \left( \frac{50}{14} \right) = 3.5714285714285716 \][/tex]

Rounding to the nearest tenth, we get:
[tex]\[ x \approx 3.6 \][/tex]

5. Calculate the y-coordinate of the treasure:

[tex]\[ y = \left( \frac{5}{5+9} \right) (15 - 0) + 0 = \left( \frac{5}{14} \right) \cdot 15 \][/tex]
[tex]\[ y = \left( \frac{75}{14} \right) = 5.357142857142857 \][/tex]

Rounding to the nearest tenth, we get:
[tex]\[ y \approx 5.4 \][/tex]

Thus, the coordinates of the treasure are:
[tex]\[ (x, y) = (3.6, 5.4) \][/tex]

Therefore, based on the given options, none of them exactly match the coordinates calculated. The rounded coordinates of the treasure are:

[tex]\[ (3.6, 5.4) \][/tex]