Question

What are the Manhattan and Pythagorean distances (in m), respectively between points A and B in the figure below, where the Euclidean distance between A and C is 4 m, and the Euclidean distance between C and B is 4 m? All the cells have the same edge lengths. 

• A. 9.0 and 5.7
• B. 5.7 and 8.0
• C. 5.7 and 5.7
• D. 8.0 and 5.7

Hint:

On grids, Manhattan distance is the $\ell_1$ norm (sum of horizontal and vertical moves). The straight-line distance is the $\ell_2$ norm via Pythagoras: $\sqrt{\Delta x^2+\Delta y^2}$.

Answer 1

The Correct Option is D

From the figure, $A$ and $C$ are opposite corners of the bottom side and $C$ and $B$ are opposite corners of the right side; both side lengths are $4$ m. Hence the square has side $4$ m.
Manhattan distance $d_M(A,B)$: sum of axis-aligned displacements $⇒ 4 + 4 = 8$ m.
Pythagorean (Euclidean) distance $d_E(A,B)$: diagonal of a $4\times 4$ square $⇒ \sqrt{4^2+4^2}=4\sqrt{2}\approx 5.656 \approx 5.7$ m (to one decimal).
\[ \boxed{d_M=8.0\ \text{m}, d_E\approx 5.7\ \text{m}} \]