Question

5. To get to the library from his house, Robert biked 6 kilometers due east and then

8 kilometers due south. On the way back, he cut across a field, taking the shortest
possible route home.
How far did Robert bike on the round-trip?
Home
6 km
8 km
Library

Answer 1

24 kilometers.

Explanation:

The shortest path between two points is a straight segment that connects the two points.

Refer to the diagram attached. The 6-km segment and the 8-km segment are normal to each other. Together with the segment that joins the library and the house, the three segments now form a right triangle.

• The two shorter segments are the two legs, and
• The longer segment that joins the library and the house is the hypotenuse.

The length of the hypotenuse can be found with the Pythagorean Theorem.

`\begin{aligned}\text{Hypotenuse} &= \sqrt{(\text{Leg 1})^(2) + (\text{Leg 2})^(2)}\\&= \sqrt{6^(2) + 8^(2)}\\&= √(36 + 64) \\&= √(100)\\&= \rm 10\;km\end{aligned}`.

The length of the round-trip will equal to the sum of the length of the three segments:
`\rm 6\;km + 8\;km + 10\;km = 24\;km`.