Fatima starts from point P, goes North for 3 km, and then East for 4 km to reach point Q. The direct distance (in km) from point P to point Q is _________.
Options:
(A) 3
(B) 4
(C) 5
(D) 7
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.59)**
**1. Intuitive Physical Setup & Given Parameters**:
- Displacement 1: 3 km North ($+y$ direction).
- Displacement 2: 4 km East ($+x$ direction).
- North and East are orthogonal ($90^\circ$ angle).
**2. Step-by-Step Mathematical Derivation**:
1. **Right-Angled Triangle Geometry**:
The two displacements form the legs of a right triangle:
- Base = 4 km
- Height = 3 km
2. **Pythagorean Theorem**:
$d = \sqrt{\text{Base}^2 + \text{Height}^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5\text{ km}$
The direct displacement from point P to Q is exactly **5 km**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** 3: ❌ Incorrect.
- **(B)** 4: ❌ Incorrect.
- **(C)** 5: ✅ **CORRECT**. Canonical 3-4-5 Pythagorean right triangle.
- **(D)** 7: ❌ Incorrect. Manhattan distance (scalar sum $3 + 4$).
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **C**
- **Exam Rule**: Direct straight-line distance is Euclidean norm: $d = \sqrt{\Delta x^2 + \Delta y^2}$.
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