Simon draws a hexagon. His hexagon has rotational symmetry of order two, two lines of symmetry, three pairs of parallel sides, and a perimeter of 32 centimeters. Draw a sketch that could be the same as Simon’s hexagon. Label the lengths of each side.
If we consider a regular hexagon, that is a hexagon where all sides and all angles are equal, we know that it has three pairs of parallel sides. It meets one of the requirements for Simon’s hexagon. However, when it comes to rotational symmetry, this regular hexagon has an order of six. To think about rotational symmetry, imagine putting your finger on the yellow dot in the middle of this hexagon and then spinning. Rotational symmetry of order six means that there are six ways you could spin this hexagon. And it would look the same. We’re looking for a hexagon that has rotational symmetry of order two where there are only two places where the shape would look the same if you were spinning it.
If we take a regular hexagon and stretch it in width, we keep the three pairs of parallel sides. But this shape no longer has a rotational symmetry of order six. If you put your finger in the middle of the shape and spin it 180 degrees, it would look the same which means this shape does have rotational symmetry of order two. Now, we need to sketch some lines of symmetry. These are the places we could fold our shape in half. And all the edges would line up evenly. Heres one line of symmetry and another line of symmetry. Now that we’ve shown this shape has a rotational symmetry of order two, two lines of symmetry, and three pairs of parallel sides, we need to label it so that it has a perimeter of 32 centimeters.
The perimeter of a hexagon is equal to adding up all six side lengths. And if the perimeter equals 32, all the sides need to add up to 32. In this hexagon, we know that the top and the bottom must be equal to each other. So let’s say that there are 10 centimeters each. Since the perimeter must measure 32 and we’ve used 20 of those centimeters on the top and bottom, this means that the other four sides must be equal to 12 centimeters when theyre added together. Remember how we started with a regular hexagon and we just stretched its width. This means we know the remaining four sides must be equal to each other. And so we divide 12 by four and we get three. The other four sides need to measure three centimeters. And we know that 32 does equal three plus three plus three plus three plus 10 plus 10. And so our hexagon fits all the requirements.
This is not the only possible option for a correct hexagon. We could take our regular hexagon and compress it. It has three pairs of parallel sides, rotational symmetry of order two, two lines of symmetry. Now, we need to give it a perimeter of 32. If we make the top two centimeters, the bottom equals two centimeters. This means that out of our 32, weve used four. And we have 28 remaining centimeters to divide evenly into our four sides. 28 divided by four is seven. If we label the remaining four sides as seven centimeters, then this hexagon also fits the requirements.
Here is our third option that is a less common hexagon that could still fit all the criteria. It has six sides, three sets of parallel sides, rotational symmetry of order two, and two lines of symmetry. In this case, we could label all the small sides four centimeters and the two longer sides eight centimeters. I know that four times four equals 16. And eight times two equals 16. 16 plus 16 equals 32. And so this third hexagon does have a perimeter of 32 centimeters. You could draw any hexagon as long as it meets these four criteria.