A treasure map says that a treasure is buried a distance between a rock and a tree in a 5:1 ratio. Onto a coordinate plane to find the exact location use the following 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 coordinates to the nearest tenth?
(5.7, 7.5)
(11.4, 14.2)
Answer (1)
Substitute the given values into the formula for the x-coordinate: x = ( 5 + 1 5 ) ( 11.4 − 5.7 ) + 5.7 = 10.45 .
Substitute the given values into the formula for the y-coordinate: y = ( 5 + 1 5 ) ( 14.2 − 7.5 ) + 7.5 = 13.08333... .
Round the x and y coordinates to the nearest tenth: x = 10.5 and y = 13.1 .
State the coordinates of the treasure: ( 10.5 , 13.1 ) .
Explanation
Problem Analysis and Given Data We are given the coordinates of a rock ( x 1 , y 1 ) = ( 5.7 , 7.5 ) and a tree ( x 2 , y 2 ) = ( 11.4 , 14.2 ) . The treasure is buried on the line segment connecting the rock and the tree, dividing the distance in a ratio of m : n = 5 : 1 . We are also given the formulas to find the coordinates of the treasure:
x = ( m + n m ) ( x 2 − x 1 ) + x 1 y = ( m + n m ) ( y 2 − y 1 ) + y 1
Our goal is to find the coordinates ( x , y ) of the treasure to the nearest tenth.
Calculating the x-coordinate First, let's substitute the given values into the formula for x :
x = ( 5 + 1 5 ) ( 11.4 − 5.7 ) + 5.7 x = ( 6 5 ) ( 5.7 ) + 5.7 x = 6 5 × 5.7 + 5.7 x = 6 28.5 + 5.7 x = 4.75 + 5.7 x = 10.45
Rounding to the nearest tenth, we get x = 10.5 .
Calculating the y-coordinate Next, let's substitute the given values into the formula for y :
y = ( 5 + 1 5 ) ( 14.2 − 7.5 ) + 7.5 y = ( 6 5 ) ( 6.7 ) + 7.5 y = 6 5 × 6.7 + 7.5 y = 6 33.5 + 7.5 y = 5.58333... + 7.5 y = 13.08333...
Rounding to the nearest tenth, we get y = 13.1 .
Final Answer Therefore, the coordinates of the treasure are ( 10.5 , 13.1 ) to the nearest tenth.
Examples
Imagine you're designing an escape room and need to place a clue at a specific point between two landmarks. Using the section formula, you can accurately calculate the clue's coordinates based on a given ratio, ensuring players find it at the intended location. This method is also used in computer graphics to draw lines and curves, and in GPS navigation to calculate intermediate points along a route.
