Question

A person travels through 5 cities, A, B, C, D and E. City E is 2 km west of D. D is 3 km north-east of A. C is 5 km north of B and 4 km west of A. If this person visits these cities in the sequence B - C - A - E - D, what is the effective distance covered between cities B and D?

• A. 13 km
• B. 9 km
• C. 10 km
• D. 11 km

Hint:

Place A at the origin and work out each city's position using the given directions (north, west, north-east), then add up the lengths of the four legs of the route B-C-A-E-D.

Answer 1

The Correct Option is A

Step 1: Set up a coordinate system.
Take east as the positive x direction and north as the positive y direction, with city A at the origin, so \(A=(0,0)\).

Step 2: Place C and B using the given directions.
C is 4 km west of A, a pure horizontal move, so \(C=(-4,0)\).
C is 5 km north of B, a pure vertical move, so B sits 5 km directly south of C: \(B=(-4,-5)\).
This also confirms that the distance from B to C is exactly 5 km, and from C to A is exactly 4 km, matching the figures given in the question.

Step 3: Place D using the north-east direction from A.
North-east sits exactly halfway between north and east, at \(45^{\circ}\) to both axes.
So D, 3 km north-east of A, has coordinates
\[ D = (3\cos 45^{\circ},\ 3\sin 45^{\circ}) \approx (2.12,\ 2.12) \]

Step 4: Place E using its position relative to D.
E is 2 km west of D, a pure horizontal move, so
\[ E = (2.12-2,\ 2.12) = (0.12,\ 2.12) \]

Step 5: Add up the distance travelled along the actual route B - C - A - E - D.
Distance B to C \(= 5\) km (given directly).
Distance C to A \(= 4\) km (given directly).
Distance A to E, using the distance formula:
\[ AE = \sqrt{(0.12-0)^2+(2.12-0)^2} = \sqrt{0.01+4.49} \approx 2.12 \text{ km} \]
Distance E to D \(= 2\) km (given directly).

Step 6: Total the segments.
\[ 5 + 4 + 2.12 + 2 = 13.12 \approx 13 \text{ km} \]

Step 7: Check the other options.
Options (2), (3) and (4), that is 9 km, 10 km and 11 km, are all smaller than the actual distance covered by the route through C, A and E, so they undercount the journey.
Only 13 km matches the sum of all four legs of the trip.

Final Answer:
The effective distance covered between city B and city D, travelling through C, A and E, is about 13 km. \[ \boxed{13 \text{ km}} \]