A regular hexagon can be divided into six congruent equilateral triangles. A.On the paper where you constructed your regular hexagon inscribed in a circle, use your straightedge to draw three different line segments that divide your constructed hexagon into six equilateral triangles. B.Explain how you know that each of these triangles is equilateral.

Question

GPT 3.5 · Accepted Answer

To construct three line segments that divide the hexagon into six equilateral triangles, we can start by connecting the midpoints of alternating sides of the hexagon.

First, connect the midpoint of one side to the midpoint of the side two spaces away (e.g., if we label the sides of the hexagon 1, 2, 3, 4, 5, 6, then we connect the midpoint of side 1 to the midpoint of side 4). This line segment will divide the hexagon into two congruent equilateral triangles.

Next, connect the midpoint of another side to the midpoint of the side three spaces away (e.g., connect the midpoint of side 2 to the midpoint of side 5). This line segment will also divide the hexagon into two congruent equilateral triangles.

Finally, connect the midpoint of the remaining side to the midpoint of the side four spaces away (e.g., connect the midpoint of side 3 to the midpoint of side 6). This line segment will divide the hexagon into the last two congruent equilateral triangles.

To explain why each of these triangles is equilateral, we know that a hexagon inscribed in a circle has a special property: all of its angles are equal to 120 degrees. By connecting the midpoints of alternating sides, we are essentially connecting opposite corners of these 120-degree angles.

Since the opposite corners of a 120-degree angle are 180 degrees apart, the line segment connecting them forms a straight line. In an equilateral triangle, all angles are equal to 60 degrees, which means each side angle is 180 degrees divided by three, or 60 degrees. Thus, the line segments we drew connect opposite corners of 120-degree angles, resulting in triangles with three equal angles of 60 degrees. Therefore, each of the triangles is equilateral.