To determine the gym's location, let's break down the problem using the given coordinates and the fraction specifying the distance along the line segment connecting Ping's and Ari's homes.
1. Identify the Coordinates:
- Ping's home is at the corner of 3rd Street and 6th Avenue:
[tex]\( P(3, 6) \)[/tex]
- Ari's home is at the corner of 21st Street and 18th Avenue:
[tex]\( A(21, 18) \)[/tex]
2. Understand the Fraction:
- The gym is [tex]\(\frac{2}{3}\)[/tex] of the distance from Ping's home to Ari's home.
- This implies that [tex]\( \frac{m}{m+n} = \frac{2}{3} \)[/tex].
3. Define the General Formula:
Given the formulas for determining coordinates at a certain fraction of the distance between two points:
[tex]\[
x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1
\][/tex]
[tex]\[
y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1
\][/tex]
4. Apply the Provided Values:
- [tex]\( x_1 = 3 \)[/tex]
- [tex]\( y_1 = 6 \)[/tex]
- [tex]\( x_2 = 21 \)[/tex]
- [tex]\( y_2 = 18 \)[/tex]
- [tex]\( \frac{m}{m+n} = \frac{2}{3} \)[/tex]
5. Calculate the x-coordinate of the gym:
[tex]\[
x = \left(\frac{2}{3}\right)(21 - 3) + 3
\][/tex]
Simplifying the expression inside the parentheses:
[tex]\[
x = \left(\frac{2}{3}\right)(18) + 3
\][/tex]
Multiplying:
[tex]\[
x = 12 + 3
\][/tex]
Adding:
[tex]\[
x = 15
\][/tex]
6. Calculate the y-coordinate of the gym:
[tex]\[
y = \left(\frac{2}{3}\right)(18 - 6) + 6
\][/tex]
Simplifying the expression inside the parentheses:
[tex]\[
y = \left(\frac{2}{3}\right)(12) + 6
\][/tex]
Multiplying:
[tex]\[
y = 8 + 6
\][/tex]
Adding:
[tex]\[
y = 14
\][/tex]
7. Determine the Gym's Location:
- The coordinates of the gym are (15, 14), which corresponds to the corner of 15th Street and 14th Avenue.
Thus, the gym is located at 15th Street and 14th Avenue.
