Question

1 compound
as shown in (1), the side length of each small square is 1.

1. from \start\ to \end\, which tortoise has the shortest route?
2. the length of the shortest route in (1) is . (write the answer in its simplest form.)

Explanation:

Step1: Calculate the length of each route

For route A:
It has \(2\) horizontal segments of length \(1\) each and \(3\) vertical segments of length \(1\) each. Using the Pythagorean theorem for the non - straight parts (but we can also count the number of unit segments). The length \(L_A=\sqrt{1^{2}+1^{2}}+\sqrt{1^{2}+1^{2}} + 1+1\). But if we consider moving along the grid (counting the number of unit - length moves, for a right - angled path, the sum of the horizontal and vertical displacements). The horizontal displacement \(x = 3\) and the vertical displacement \(y=2\). The length \(L_A=3 + 2=5\).

For route B:
The horizontal displacement \(x = 4\) and the vertical displacement \(y = 1\). The length \(L_B=4+1 = 5\).

For route C:
Using the Pythagorean theorem \(a = 4\) (horizontal) and \(b = 2\) (vertical). The length \(L_C=\sqrt{4^{2}+2^{2}}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\approx4.47\).

Step2: Compare the lengths

Since \(2\sqrt{5}\approx4.47<5\), route C is the shortest.

Answer:

1. C.
2. \(2\sqrt{5}\)