In the app below you can see a [i]polygon: [/i]polygon is a word derived from Greek, and means "many angles".[br][br]A polygon can be [i]simple[/i], when its boundary does not cross itself, otherwise it's [i]self-intersecting[/i].[br][br]Drag the vertices of the polygon (green points) and create [i]simple[/i] and [i]self-intersecting[/i] figures. The intersection points will be shown in orange.[br]
When polygons are [i]simple[/i], we can characterize them further, as [i]convex [/i]or [i]concave[/i].[br][br]Explore the applet below, create a few different [i]convex [/i]and [i]concave[/i] polygons, observe the measure of the angles and the position of the diagonals with respect to the portion of the plane enclosed within its sides.[br][br]Formulate your own conjecture about how to differentiate a [i]convex [/i]and a [i]concave [/i]polygon, depending on these properties of the figure.
What can you say about the measures of the interior angles of a [i]convex [/i]polygon?[br]Do they all have a common characteristic?
In a[i] convex polygon[/i], all [i]interior angles [/i]are [i]less than or equal to[/i] 180°.
What can you say about the measures of the interior angles of a [i]concave[/i] polygon?[br]Do some of them have a common characteristic?
A [i]concave polygon[/i] has always at least [i]one reflex interior angle, [/i]i.e. an angle whose measure is between 180° and 360°.
Describe the [i]difference [/i]between the [i]position[/i] of the [i]diagonals[/i] of a [i]convex [/i]and of a [i]concave [/i]polygon, with respect to the portion of the plane enclosed within its sides.
[i]All [/i]the diagonals of a [i]convex[/i] polygon [i]lie inside[/i] the portion of the plane enclosed within its sides, while [i]some [/i]of the diagonals of a [i]concave[/i] polygon will [i]lie outside[/i] of it.