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.
$\begin{array}{l}
x=\left(\frac{m}{m+n}\right)\left(x_2-x_1\right)+x_1 \\
v=\left(\frac{m}{m+n}\right)\left(v_2-v_1\right)+v_1
\end{array}$
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 (1)
Define Ping's location as (3, 6) and Ari's location as (21, 18).
Determine the fraction of the distance: 3 2 .
Calculate the street location: x = ( 3 2 ) ( 21 − 3 ) + 3 = 15 .
Calculate the avenue location: v = ( 3 2 ) ( 18 − 6 ) + 6 = 14 .
The gym is located at 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 location of the gym, which is 3 2 the distance from Ping's home to Ari's home. We are also given the formula to find a point a fraction of the distance between two points.
Setting up the Formula 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 . Therefore, m + n m = 2 + 1 2 = 3 2 .
Calculating the Street Location Now, we can calculate the street location of the gym using the formula: x = ( m + n m ) ( x 2 − x 1 ) + x 1 x = ( 3 2 ) ( 21 − 3 ) + 3 x = ( 3 2 ) ( 18 ) + 3 x = 12 + 3 x = 15 So, the gym is located on 15th Street.
Calculating the Avenue Location Next, we calculate the avenue location of the gym using the formula: v = ( m + n m ) ( v 2 − v 1 ) + v 1 v = ( 3 2 ) ( 18 − 6 ) + 6 v = ( 3 2 ) ( 12 ) + 6 v = 8 + 6 v = 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
This formula is useful in many real-world scenarios, such as determining the location of a meeting point between two people who are starting from different locations. For example, if two friends are driving from different cities to meet at a restaurant that is a certain fraction of the distance between their cities, they can use this formula to find the exact location of the restaurant. This ensures they both travel a fair distance and arrive at the meeting point without either person having to travel significantly further than the other.
