Ping lives at the corner of 3rd Street and 6th Avenue. Ari lives at the corner of 21st Street and 18th Avenue. There is a gym $\frac{2}{3}$ the distance from Ping's home to Ari's home. $x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1$ $v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1$ Where is the gym? A. 9th Street and 10th Avenue B. 12th Street and 12th Avenue C. 14th Street and 12th Avenue D. 15th Street and 14th Avenue
Answer (2)
Define Ping's location as (3, 6) and Ari's location as (21, 18).
Calculate the fraction of the distance: 3 2 .
Use the formula x = ( 3 2 ) ( 21 − 3 ) + 3 to find the street: x = 15 .
Use the formula v = ( 3 2 ) ( 18 − 6 ) + 6 to find the avenue: v = 14 .
The gym is located at 15th Street and 14th Avenue: 15 t h St ree t an d 14 t h A v e n u e .
Explanation
Understanding the Problem We are given the coordinates of Ping's home (3rd Street, 6th Avenue) and Ari's home (21st Street, 18th Avenue). We need to find the coordinates of the gym, which is located 3 2 of the distance from Ping's home to Ari's home. We are also given the formulas to calculate the coordinates of a point that is a fraction of the distance between two points.
Setting up the Coordinates and Ratio Let Ping's location be ( x 1 , v 1 ) = ( 3 , 6 ) and Ari's location be ( x 2 , v 2 ) = ( 21 , 18 ) . The gym is 3 2 the distance from Ping to Ari, so m = 2 and n = 1 . Thus, m + n m = 2 + 1 2 = 3 2 .
Calculating the Street Location We use the formula x = ( m + n m ) ( x 2 − x 1 ) + x 1 to find the street location of the gym. Substituting the values, we get: x = 3 2 ( 21 − 3 ) + 3 = 3 2 ( 18 ) + 3 = 12 + 3 = 15 So, the gym is located on 15th Street.
Calculating the Avenue Location We use the formula v = ( m + n m ) ( v 2 − v 1 ) + v 1 to find the avenue location of the gym. Substituting the values, we get: v = 3 2 ( 18 − 6 ) + 6 = 3 2 ( 12 ) + 6 = 8 + 6 = 14 So, the gym is located on 14th Avenue.
Final Answer Therefore, the gym is located at the corner of 15th Street and 14th Avenue.
Examples
Imagine you are planning a meeting spot with a friend. You live at one location, and your friend lives at another. To make it fair, you decide to meet at a cafe that's a certain fraction of the distance between your homes. Using the given formulas, you can calculate the exact street and avenue where the cafe should be located, ensuring a balanced commute for both of you. This method is useful in logistics, urban planning, and even in everyday situations to find a convenient midpoint or a location that favors one party slightly more than the other.
The gym is located at 15th Street and 14th Avenue, which is 3 2 of the distance between Ping's home and Ari's home. This was calculated using the provided formulas for determining a point at a specific fraction between two known coordinates. Thus, the selected answer is option D: 15th Street and 14th Avenue.
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