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Sagot :
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.
[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.
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