Answer:
Explanation:
To find the possible coordinates of the library, we need to determine the line of points that are equidistant from the school and the bank, since the distance from the library to the school is the same as the distance from the bank to the school. This line will be the perpendicular bisector of the segment connecting the school and the bank.
First, we can find the midpoint of the segment connecting the school and the bank:
Midpoint = ((x1 + x2)/2, (y1 + y2)/2) = ((5 + 1)/2, (4 + 3)/2) = (3, 3.5)
Next, we can find the slope of the segment connecting the school and the bank:
Slope = (y2 - y1)/(x2 - x1) = (3 - 4)/(1 - 5) = -1/4
The slope of the perpendicular bisector will be the negative reciprocal of the slope of the segment connecting the school and the bank:
Slope of perpendicular bisector = -1/(-1/4) = 4
Now we have the midpoint and the slope of the perpendicular bisector, so we can use the point-slope form of the equation of a line to find the equation of the perpendicular bisector:
y - y1 = m(x - x1)
y - 3.5 = 4(x - 3)
y - 3.5 = 4x - 12
y = 4x - 8.5
Therefore, the possible coordinates of the library must lie on the line y = 4x - 8.5. We can check each of the answer choices to see which one lies on this line:
(1, 0): y = 4x - 8.5 = 4(1) - 8.5 = -4.5 (does not lie on the line)
(2, 2): y = 4x - 8.5 = 4(2) - 8.5 = -0.5 (does not lie on the line)
(4, 4): y = 4x - 8.5 = 4(4) - 8.5 = 7.5 (does not lie on the line)
(5, 5): y = 4x - 8.5 = 4(5) - 8.5 = 11.5 (does not lie on the line)
(6, 6): y = 4x - 8.5 = 4(6) - 8.5 = 15.5 (does not lie on the line)
(7, 7): y = 4x - 8.5 = 4(7) - 8.5 = 19.5 (does not lie on the line)