A treasure map says that a treasure is buried such 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.
[tex]x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1[/tex]
[tex]y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1[/tex]
What are the coordinates of the treasure? If necessary, round the coordinates to the nearest tenth.
A. (11.4, 14.2)
B. (7.6, 8.8)
C. (5.7, 7.5)
D. (10.2, 12.6)
Answer (1)
Use the section formula to find the coordinates of a point dividing a line segment in a given ratio.
Substitute the given coordinates of the rock (21, 22) and the tree (24, 25), and the ratio 5:9 into the section formula.
Calculate the x-coordinate: x = ( 14 5 ) ( 24 − 21 ) + 21 ≈ 22.1 .
Calculate the y-coordinate: y = ( 14 5 ) ( 25 − 22 ) + 22 ≈ 23.1 . The coordinates of the treasure are ( 22.1 , 23.1 ) .
Explanation
Problem Analysis We are given that a treasure is buried on a line segment connecting a rock and a tree, dividing the distance in a 5:9 ratio. We are also given the coordinates of the rock (21, 22) and the tree (24, 25). We need to find the coordinates of the treasure using the section formula.
Section Formula The section formula is given by: x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1 where ( x 1 , y 1 ) are the coordinates of the rock, ( x 2 , y 2 ) are the coordinates of the tree, and m : n is the ratio in which the treasure divides the distance.
Substitute Values We have ( x 1 , y 1 ) = ( 21 , 22 ) , ( x 2 , y 2 ) = ( 24 , 25 ) , and m : n = 5 : 9 . Thus, m = 5 and n = 9 . Substituting these values into the section formula, we get: x = ( 5 + 9 5 ) ( 24 − 21 ) + 21 y = ( 5 + 9 5 ) ( 25 − 22 ) + 22
Calculate x-coordinate Now, we calculate the x-coordinate: x = ( 14 5 ) ( 3 ) + 21 x = 14 15 + 21 x = 14 15 + 14 294 x = 14 309 x ≈ 22.0714
Calculate y-coordinate Next, we calculate the y-coordinate: y = ( 14 5 ) ( 3 ) + 22 y = 14 15 + 22 y = 14 15 + 14 308 y = 14 323 y ≈ 23.0714
Round to Nearest Tenth Rounding the coordinates to the nearest tenth, we get: x ≈ 22.1 y ≈ 23.1 Therefore, the coordinates of the treasure are approximately (22.1, 23.1).
Final Answer The coordinates of the treasure are approximately (22.1, 23.1).
Examples
The section formula is useful in various real-world scenarios, such as determining the location of a point dividing a line segment in a specific ratio. For instance, in urban planning, if a new facility needs to be built between two existing locations, the section formula can help determine the exact coordinates of the new facility based on desired proximity ratios to the existing locations. This ensures optimal placement and accessibility for the community.
