Tia lives at the corner of 4th Street and 8th Avenue. Lei lives at the corner of 12th Street and 20th Avenue. The fruit market is $\frac{3}{4}$ the distance from Tia's home to Lei's home.
$\begin{array}{l}
x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \\
y=\left(\frac{m}{m+n}\right)\left(y_2-y_1\right)+y_1
\end{array}$
Where is the fruit market?
A. 6th Street and 11th Avenue
B. 10th Street and 17th Avenue
C. 9th Street and 15th Avenue
D. 8th Street and 14th Avenue
Answer (2)
Define Tia's and Lei's locations as coordinates (4, 8) and (12, 20), respectively.
Determine the ratio m + n m as 4 3 , representing the fruit market's position along the line segment.
Calculate the x-coordinate of the fruit market using the formula: x = ( 4 3 ) ( 12 − 4 ) + 4 = 10 .
Calculate the y-coordinate of the fruit market using the formula: y = ( 4 3 ) ( 20 − 8 ) + 8 = 17 . The fruit market is at 10 t h St ree t an d 17 t h A v e n u e .
Explanation
Problem Analysis Let's analyze the problem. We are given the coordinates of Tia's and Lei's homes, and we know that the fruit market is located 4 3 of the distance from Tia's home to Lei's home. We are also given the formulas to calculate the coordinates of a point that divides a line segment in a given ratio.
Define Coordinates and Ratio Tia's location is ( x 1 , y 1 ) = ( 4 , 8 ) and Lei's location is ( x 2 , y 2 ) = ( 12 , 20 ) . The fruit market is 4 3 the distance from Tia's to Lei's, so the ratio is m : n = 3 : 1 . Thus, m + n m = 3 + 1 3 = 4 3 .
Calculate x-coordinate Now, we use the given formula to find the x-coordinate of the fruit market: x = ( m + n m ) ( x 2 − x 1 ) + x 1 x = ( 4 3 ) ( 12 − 4 ) + 4 x = ( 4 3 ) ( 8 ) + 4 x = 6 + 4 x = 10
Calculate y-coordinate Next, we use the given formula to find the y-coordinate of the fruit market: y = ( m + n m ) ( y 2 − y 1 ) + y 1 y = ( 4 3 ) ( 20 − 8 ) + 8 y = ( 4 3 ) ( 12 ) + 8 y = 9 + 8 y = 17
Final Answer Therefore, the location of the fruit market is (10, 17), which corresponds to 10th Street and 17th Avenue.
Examples
This type of problem is useful in urban planning and logistics. For example, if you need to place a service or facility (like a bus stop or a store) at a certain proportion along a route between two points, you can use these formulas to determine the exact location. This ensures that the service is conveniently located for the population it serves, optimizing accessibility and usage.
The fruit market is located at 10th Street and 17th Avenue, determined by calculating its position as 4 3 of the way from Tia's home to Lei's home using the given coordinates. The final answer is based on the calculated coordinates (10, 17).
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