Question

Instead of turning right at the end if he took left and walked 20 km, what is the shortest distance to his starting point?

• A. \(3\sqrt{7}\) km
• B. \(2\sqrt{5}\) km
• C. \(7\sqrt{2}\) km
• D. \(5\sqrt{2}\) km

Hint:

Translate every turn into a direction on the \(xy\)-plane and keep running coordinates. Then use the Pythagorean theorem for the straight-line distance.

Answer 1

The Correct Option is D

Path with coordinates:
Start at \(O(0,0)\). Take \(+x\) east, \(+y\) north.
- Walk 25 km west \(\Rightarrow (-25,0)\).
- Left (to south) 15 km \(\Rightarrow (-25,-15)\).
- Left (to east) 30 km \(\Rightarrow (5,-15)\).
- Now take left (to north) and walk 20 km \(\Rightarrow (5,5)\).
Distance from start:
\[ d=\sqrt{(5-0)^2+(5-0)^2}=\sqrt{25+25}=\sqrt{50}=5\sqrt{2}\ \text{km}. \] \[ \boxed{5\sqrt{2}\ \text{km}} \]