Question

explore and show straight line→shortest

1. as shown below, the edge length of the cube is 1. a ladybug crawl from vertex p to vertex q along the

surface of the cube. three different colors represent three different routes. point d is the midpoint of \\( \overline { a b } \\).
(1) unfold the left and front faces of the cube and draw the other two routes.
(2) which path is the shortest?
a. \\( p \
ightarrow a \
ightarrow b \
ightarrow q \\) b. \\( p \
ightarrow c \
ightarrow q \\) c. \\( p \
ightarrow d \
ightarrow q \\)
(3) the length of the shortest path in (2) is _.
a. 2.9 b. \\( \sqrt { 5 } \\) c. 3

Explanation:

Step1: Calculate the length of path \(P

ightarrow A
ightarrow B
ightarrow Q\)
The length of this path is \(PA + AB+BQ\). Since the edge - length of the cube is \(1\), \(PA = 1\), \(AB = 1\), \(BQ = 1\). So the length \(L_1=1 + 1+1=3\).

Step2: Calculate the length of path \(P

ightarrow C
ightarrow Q\)
Unfold the left and top - front faces. The horizontal distance is \(1 + 0.5=1.5\) and the vertical distance is \(1 + 0.5 = 1.5\). Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 1.5\), \(b = 1.5\). Then \(L_2=\sqrt{(1 + 0.5)^{2}+(1 + 0.5)^{2}}=\sqrt{2\times(1.5)^{2}}=\sqrt{4.5}\approx2.12\)

Step3: Calculate the length of path \(P

ightarrow D
ightarrow Q\)
Unfold the left and front faces. The horizontal distance is \(1+1 = 2\) and the vertical distance is \(1\). Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 2\), \(b = 1\). Then \(L_3=\sqrt{(1 + 1)^{2}+1^{2}}=\sqrt{4 + 1}=\sqrt{5}\approx2.24\)

Answer:

(2) The shortest path is \(P
ightarrow C
ightarrow Q\) (Option B).
(3) The length of the shortest path is \(\sqrt{5}\) (Option B).