Q. There are 7 places A, B, C, D, E, F and G in a city connected by various roads AB, AC, CD, DE, BF, EG and FG. A is 6 km south of B. A is 10 km west of C. D is 5 km east of E. C is 6 km north of D. F is 9 km west of B. F is 12 km north of G. A person travels from D to F through these roads. What is the distance covered by the person?

Explanation : 

1. Assign convenient coordinates to each place, taking east as the +x-direction and north as the +y-direction. For example, let A = (0, 0).
– A is 6 km south of B ⇒ B = (0, 6)
– A is 10 km west of C ⇒ C = (10, 0)
– C is 6 km north of D ⇒ D = (10, –6)
– D is 5 km east of E ⇒ E = (5, –6)
– F is 9 km west of B ⇒ F = (–9, 6)
– F is 12 km north of G ⇒ G = (–9, –6)

2. Compute the lengths of the roads (all align with the axes):
– AB = distance between (0,0) and (0,6) = 6
– AC = distance between (0,0) and (10,0) = 10
– CD = distance between (10,0) and (10,–6) = 6
– DE = distance between (10,–6) and (5,–6) = 5
– BF = distance between (0,6) and (–9,6) = 9
– EG = distance between (5,–6) and (–9,–6) = 14
– FG = distance between (–9,6) and (–9,–6) = 12

3. Two possible routes from D to F along the given roads both give the same total:
• D → C → A → B → F
= DC + CA + AB + BF
= 6 + 10 + 6 + 9
= 31 km

• D → E → G → F
= DE + EG + GF
= 5 + 14 + 12
= 31 km

Hence the distance traveled is 31 km.