A treasure map says 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.
$x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 $
$y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1$
What are the coordinates of the treasure? If necessary, round the coordinates to the nearest tenth.
Answer (2)
The treasure is located on the line segment connecting the rock at (11, 12) and the tree at (14, 15).
The treasure divides the distance between the rock and the tree in a 5:9 ratio.
Applying the section formula, the coordinates of the treasure are calculated as x = 14 169 ≈ 12.0714 and y = 14 183 ≈ 13.0714 .
Rounding to the nearest tenth, the coordinates of the treasure are ( 12.1 , 13.1 ) .
Explanation
Problem Analysis and Given Data We are given the coordinates of a rock ( x 1 , y 1 ) = ( 11 , 12 ) and a tree ( x 2 , y 2 ) = ( 14 , 15 ) . The treasure is buried along the line segment connecting these two points, dividing the distance in a ratio of 5 : 9 . We need to find the coordinates of the treasure using the section formula.
Stating the Section Formula The section formula for a point dividing a line segment in the ratio m : n is given by:
x = m + n m x 2 + n x 1 y = m + n m y 2 + n y 1
In our case, m = 5 , n = 9 , x 1 = 11 , y 1 = 12 , x 2 = 14 , and y 2 = 15 .
Applying the Formula Substituting the given values into the section formula, we get:
x = 5 + 9 5 ( 14 ) + 9 ( 11 ) = 14 70 + 99 = 14 169 ≈ 12.0714 y = 5 + 9 5 ( 15 ) + 9 ( 12 ) = 14 75 + 108 = 14 183 ≈ 13.0714
Rounding to Nearest Tenth Rounding the coordinates to the nearest tenth, we have:
x ≈ 12.1 y ≈ 13.1
Final Answer Therefore, the coordinates of the treasure are approximately ( 12.1 , 13.1 ) .
Examples
Imagine you're designing a new park with a straight walking path connecting a playground and a picnic area. You want to place a water fountain along this path such that it's closer to the playground than the picnic area, specifically dividing the path in a 2:3 ratio. Using the section formula, you can determine the exact coordinates for the fountain, ensuring it's perfectly positioned to serve both areas effectively. This ensures optimal accessibility and convenience for park visitors.
The coordinates of the treasure, dividing the distance between a rock at (11, 12) and a tree at (14, 15) in a 5:9 ratio, are approximately (12.1, 13.1). We used the section formula to calculate these coordinates accurately. Rounding the results gives the final values needed to locate the treasure.
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