Campus First-Aid Station on the Coordinate Grid
On a school map, coordinates are in units of 10 m. The gate is at G(0,0), the library at L(3,4), the science lab at S(9,3), and the canteen at C(6,−3). The administration wants to place a first-aid station on the x-axis so that it is the same distance from the gate as from the library. A second proposal places a water point at the midpoint of the lab–canteen path. A student claims that the library, lab, and canteen lie on one straight line, so a single paved path could connect all three; the gardener doubts this because the points 'look scattered'. The map is used for an emergency drill, so distances and straight-line routes must be justified, not estimated from the drawing.
1. Find the coordinates of the first-aid station on the x-axis that is equidistant from the gate and the library. Convert its distance from the gate to metres.
Read the worked answer
Let the station be P(x, 0). Equidistance means PG² = PL².
PG² = x². PL² = (x − 3)² + (0 − 4)² = (x − 3)² + 16.
Set equal: x² = x² − 6x + 9 + 16 0 = −6x + 25 x = 25/6 ≈ 4.17.
So P = (25/6, 0). Each map unit is 10 m, so the distance from the gate is (25/6) × 10 = 250/6 ≈ 41.7 m.
2. Find the midpoint of the lab–canteen path and the distance from the first-aid station to that water point.
Read the worked answer
Midpoint M of S(9,3) and C(6,−3): M = ((9 + 6)/2, (3 + (−3))/2) = (15/2, 0) = (7.5, 0).
The first-aid station is at x = 25/6. Both points lie on the x-axis, so the distance is: 7.5 − 25/6 = 45/6 − 25/6 = 20/6 = 10/3 map units.
In metres: (10/3) × 10 = 100/3 ≈ 33.3 m.
3. Test the student's collinearity claim for L(3,4), S(9,3), and C(6,−3). Decide whether one straight path can connect all three.
Read the worked answer
Compare slopes.
Slope LS = (3 − 4)/(9 − 3) = −1/6. Slope LC = (−3 − 4)/(6 − 3) = −7/3.
These are not equal, so L, S, and C are not collinear. Equivalently, the area of triangle LSC is nonzero, so one straight path cannot pass through all three points. The student's claim is false; the gardener's doubt is justified.