Renuka started walking from her house. She first walked 3 km towards the west,…

Concept: When a path consists of two movements along perpendicular directions (such as West and North), the straight-line distance between the start and end points is the hypotenuse of the right triangle formed by those two movements, found via the Pythagorean theorem: c2 = a2 + b2, where c is the hypotenuse and a, b are the two legs. The overall direction from start to end combines both cardinal components — for example, a westward leg together with a northward leg gives a resultant direction of North-West.

1. Let O be Renuka's house. She walks 3 km West to point A, then 4 km North to point B, so OA = 3 km, AB = 4 km, and since West and North are perpendicular, triangle OAB is right-angled at A.

2. Apply the Pythagorean theorem to find OB: OB2 = OA2 + AB2 = 32 + 42 = 9 + 16 = 25.

3. Take the square root: OB = √25 = 5 km.

4. Determine the direction: since B lies west and north of O together, the direction from the house to Renuka's final position is North-West.

Cross-check: 3, 4, 5 form the well-known Pythagorean triple, so OB = 5 km is consistent without further computation; and since neither the westward nor the northward component is zero, the resultant direction must be a compound one (North-West) rather than a single cardinal direction.

So, Renuka is 5 km from her house, in the North-West direction.