If 𝐴𝐵𝐶𝐷𝐸𝐹 is a regular hexagon whose geometrical center is 𝑁, which of the following directed segments are not equivalent? Is it (A) 𝐵𝐶 and 𝐴𝑁, (B) 𝐵𝐶 and 𝑁𝐷, (C) 𝐵𝐶 and 𝑁𝐹, or (D) 𝐵𝐶 and 𝐹𝐸?
We will begin by sketching a regular hexagon 𝐴𝐵𝐶𝐷𝐸𝐹. We are told that the geometrical center of the hexagon is 𝑁 as shown on the diagram. For two directed segments to be equivalent, they need to be of equal magnitude or length and have the same direction. The directed segment 𝐵𝐶 is in all four of our options. There are three other directed segments on the hexagon that are equal to this.
Firstly, we have the directed segment 𝐹𝐸. 𝐴𝑁 is also equal to both of these as it is of equal magnitude and direction. Finally, we have the directed segment 𝑁𝐷. As 𝐵𝐶 is equal to 𝐴𝑁, option (A) is not the correct answer. 𝐵𝐶 is equal to 𝑁𝐷, so option (B) is not the correct answer. Likewise, 𝐵𝐶 is equal to 𝐹𝐸, so we can also rule out option (D). The line segment 𝑁𝐹, however, is not equal to 𝐵𝐶. As our hexagon is regular, these segments do have the same magnitude or length. However, they do not act in the same direction.
The correct answer is option (C). The directed segments 𝐵𝐶 and 𝑁𝐹 are not equivalent. 𝑁𝐹 would be equivalent to 𝐵𝐴, 𝐷𝐸, and 𝐶𝑁. Likewise, the directed segments 𝐹𝐴, 𝐸𝑁, 𝑁𝐵, and 𝐷𝐶 are equivalent as they have the same length and direction.