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

Here is another way to reach the same total, using triangle geometry instead of plotting coordinates.

From the question, B to C is 5 km (since C is 5 km due north of B) and C to A is 4 km (since C is 4 km due west of A). Both of these legs are given directly, with no computation needed, because "north of" and "west of" describe movement along a single direction only.

The tricky part is finding the leg from A to E, since neither distance is given directly. We know two things about the small triangle formed by A, D and E: the distance from A to D is 3 km (D is north-east of A), and the distance from D to E is 2 km (E is due west of D).

To use these two sides, we need the angle between them at vertex D. The direction from D back to A is south-west (the exact opposite of north-east), and the direction from D to E is due west. The angle between south-west and due west is \(45^{\circ}\), since south-west sits exactly halfway between south and west.

Now apply the law of cosines in triangle ADE, with \(DA=3\), \(DE=2\), and the included angle \(45^{\circ}\):

\[ AE^2 = DA^2 + DE^2 - 2\cdot DA\cdot DE\cdot\cos 45^{\circ} \] \[ AE^2 = 9 + 4 - 2(3)(2)(0.71) = 13 - 8.49 = 4.51 \] \[ AE \approx 2.12 \text{ km} \]

This matches the coordinate calculation exactly. Now add up the full route the person actually travels: B to C is 5 km, C to A is 4 km, A to E is about 2.12 km, and E to D is 2 km:

\[ 5+4+2.12+2 = 13.12 \approx 13 \text{ km} \]

The other options, 9 km, 10 km and 11 km, are all too small to account for every leg of the trip through C, A and E, so 13 km is the only figure that matches the full route.

\[ \boxed{13 \text{ km}} \]