Question

the library is at point (1,1). the gym is at point (1, - 4). the chemistry lab is at point (-3,6). the restrooms are at point (-3,1). the teachers lounge is at point (4,6). the principals office is at point (-5,-4). the auditorium is at point (4,-2). and the physics lab is at point (-1,-2). which of these pairs of places lie 4 units from one another?
a. the auditorium and the physics lab
b. the library and the gym
c. the library and the restrooms
d. the chemistry lab and the teachers lounge

Explanation:

Step1: Recall distance formula for points on same x - or y - axis

If two points \((x_1,y_1)\) and \((x_2,y_2)\) have \(x_1 = x_2\), the distance \(d=\vert y_1 - y_2\vert\). If \(y_1=y_2\), the distance \(d = \vert x_1 - x_2\vert\).

Step2: Check option A

Auditorium \((4,-2)\) and Physics lab \((-1,-2)\). Since \(y\) - coordinates are the same (\(y=-2\)), \(d=\vert4-(-1)\vert=\vert4 + 1\vert=5\).

Step3: Check option B

Library \((1,1)\) and Gym \((1,-4)\). Since \(x\) - coordinates are the same (\(x = 1\)), \(d=\vert1-(-4)\vert=\vert1 + 4\vert=5\).

Step4: Check option C

Library \((1,1)\) and Restrooms \((-3,1)\). Since \(y\) - coordinates are the same (\(y = 1\)), \(d=\vert1-(-3)\vert=\vert1+3\vert = 4\).

Step5: Check option D

Chemistry lab \((-3,6)\) and Teacher's lounge \((4,6)\). Since \(y\) - coordinates are the same (\(y = 6\)), \(d=\vert-3 - 4\vert=\vert-7\vert=7\).

Answer:

C. the library and the restrooms