Like a rhombus, an equilateral triangle has congruent sides. Show and describe how you might locate the third vertex point on an equilateral triangle, given [math]ST[/math] as one side of the equilateral triangle.
Because [b]regular polygons[/b] have rotational symmetry, they can be [b]inscribed [/b]in a circle. [br][br]The circumscribed circle has its center at the center of rotation and passes through all of the vertices of the regular polygon.[br][br]We might begin constructing a hexagon by noticing that a hexagon can be decomposed into six congruent equilateral triangles, formed by three of its lines of symmetry.
1.[list=a][*]Sketch a diagram of such a decomposition.[/*][*]Based on your sketch, where is the center of the circle that would circumscribe the hexagon?[/*][*]Use Geogebra tools to draw the circle that would circumscribe the hexagon.[br][/*][/list]
[b]CONSTRUCTING A REGULAR HEXAGON INSCRIBED IN A CIRCLE[/b]
2. The six vertices of the regular hexagon lie on the circle in which the regular hexagon is inscribed. The six sides of the hexagon are [b]chords[/b] of the circle. How are the lengths of these chords related to the lengths of the radii from the center of the circle to the vertices of the hexagon? That is, how do you know that the six triangles formed by drawing the three lines of symmetry are equilateral triangles? (Hint: Considering angles of rotation, can you convince yourself that these six triangles are equiangular and therefore equilateral?)
3. Based on this analysis of the regular hexagon and its circumscribed circle, illustrate and describe a process for constructing a hexagon inscribed in the given circle.
4. Modify your work with the hexagon to construct an equilateral triangle inscribed in the given circle.
[b]CONSTRUCTING A PARALLEL LINE THROUGH A GIVEN POINT[/b][br][br]5. It is often useful to be able to construct a line parallel to a given line through a point. For example, suppose we want to construct a line parallel to [math]AB[/math] thought point [math]C[/math] on the diagram below. Since we have observed that parallel lines have the same slope, the line through point [math]C[/math] will be parallel to [math]AB[/math] only if the angle formed by the line and [math]BC[/math] is congruent to angle [math]ABC[/math]. Can you describe and illustrate a strategy that will construct an angle with the vertex at point [math]C[/math] and a side parallel to [math]AB[/math]?
Strategies for constructing geometric figures:
What can we do to construct geometric figures?