Sure, let's break down the problem step by step and determine the correct coordinates of the gym.
1. Identify Given Points:
- Ping's coordinates are [tex]\((3, 6)\)[/tex]:
[tex]\[
x_1 = 3, \quad y_1 = 6
\][/tex]
- Ari's coordinates are [tex]\((21, 18)\)[/tex]:
[tex]\[
x_2 = 21, \quad y_2 = 18
\][/tex]
2. Distance Ratio:
- The gym is [tex]\(\frac{2}{3}\)[/tex] of the distance from Ping's home to Ari's home.
- Let [tex]\(m = 2\)[/tex] and [tex]\(n = 3 - 2 = 1\)[/tex].
3. Apply the Formula for the Coordinates:
- The formula for finding a point that divides the line segment between [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] is:
[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]
- Plugging in the given values:
[tex]\[
x = \left(\frac{2}{2+1}\right)(21 - 3) + 3
\][/tex]
[tex]\[
x = \left(\frac{2}{3}\right) \times 18 + 3
\][/tex]
[tex]\[
x = 12 + 3 = 15
\][/tex]
Similarly for [tex]\(y\)[/tex]:
[tex]\[
y = \left(\frac{2}{2+1}\right)(18 - 6) + 6
\][/tex]
[tex]\[
y = \left(\frac{2}{3}\right) \times 12 + 6
\][/tex]
[tex]\[
y = 8 + 6 = 14
\][/tex]
4. Conclusion:
- The coordinates of the gym are [tex]\((15, 14)\)[/tex].
Hence, the gym is located at the corner of 15th Street and 14th Avenue.
