The assistant principal drew a diagram of the courtyard on a coordinate grid. He drew the bike rack at (-6, 1), the bench at (5, 1), and a table at (2, 5). The length of each square on the grid represented one yard.
Using this coordinate grid, we can see that the bike rack is 6 units to the left and 1 unit up from the origin, the bench is 5 units to the right and 1 unit up from the origin, and the table is 2 units to the right and 5 units up from the origin.
Therefore, if each square on the grid represents one yard, we can determine the relative distances between these objects.
The distance between the bike rack and the bench can be calculated using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
d = √((5 - (-6))^2 + (1 - 1)^2)
d = √(11^2 + 0)
d = √121
d = 11 yards
Therefore, the bike rack and the bench are 11 yards apart.
Similarly, we can calculate the distance between the bike rack and the table:
d = √((2 - (-6))^2 + (5 - 1)^2)
d = √(8^2 + 4^2)
d = √(64 + 16)
d = √80
d = 8√5 yards
Therefore, the bike rack and the table are 8√5 yards apart.
Finally, we can calculate the distance between the bench and the table:
d = √((2 - 5)^2 + (5 - 1)^2)
d = √((-3)^2 + 4^2)
d = √(9 + 16)
d = √25
d = 5 yards
Therefore, the bench and the table are 5 yards apart.