Saturday, October 16, 2010

Convexity and Angles of Polygon

Polygons which look like those in the top row of Figure 4.5 we will call convex. Thus we define a polygon to be convex if it has the following property:

Given two points X and Y on the sides of the polygon, then the segment XY is wholly contained in the polygonal region surrounded by the polygon (including the polygon itself).

Observe how this condition fails in a polygon such as one chosen from the lower row in Figure 4.5 on Chapter Basic Idea of Polygon
4.6 Example polygon


You might want to go back to Figure 4.5 and verify that this condition does hold on each polygon in the top row.

Throughout this book we shall only be dealing with convex polygons, as they are generally more interesting. Consequently, to simplify our language, we shall always assume that a polygon is convex, and not say so explicitly every time.

In a polygon, let PQ and QM be two sides with common endpoint Q. Then the polygon lies within one of the two angles determined by the rays RQP and RQM. This angle is called one of the angles of the polygon. Observe that this angle has less than 180°, as illustrated:
4.7 Angles of polygon

Experiment 4.2.
  1. Besides the number of sides, two characteristics of polygons are the lengths of its sides and the measures of its angles.
  2. What do we call a quadrilateral which has four sides of the same length and which has four angles with the same measure?
  3. Can you think of a quadrilateral which has four angles with equal measures but whose sides do not all have the same length? Draw a picture. What do we call such a quadrilateral?
  4. Can you draw a quadrilateral which has four sides of equal length, but whose angles do not have the same measure?
  5. What do we call a 3-gon which has equal length sides and equal measure angles?
  6. With a ruler, draw an arbitrary looking convex quadrilateral. Measureeach of its four angles, and add these measures. Repeat with two or three other quadrilaterals.
  7. Repeat the procedure given in Part 2 with a few pentagons, and then a few hexagons.
  8. What can you conclude? Can you say what the sum of the measures of the angles of a 7-gon would be? How about a 13-gon?
  9. For the rest of this experiment, we will develop a formula to answer these questions.

Consider a convex quadrilateral. A line segment between two opposite vertices is called a diagonal. We can decompose the quadrilateral ("break it down") into two triangles by drawing a diagonal, as shown:
4.8 A diagonal

Notice that the angles of the two triangles make up the angles of the polygon. What is the sum of the angles in each triangle? In the two triangles added together? And in the polygon?

Now look at a convex pentagon. We can decompose it into triangles, using the "diagonals" from a single vertex, as shown:
4.9 Diagonals

We see that in a 5-gon we get three such triangles. Again, the angles of the triangles make up the angles of the polygon when it is decomposed in this way. What is the sum of the measures of all the angles in the triangles? What is the sum of the measures of all the angles in the polygon?

Repeat this procedure with a hexagon to find the sum of the measures of its angle. Continue the process until you can state a formula which will give the sum of the measures of the angles of an n-gon in terms of n. If you have succeeded, you will have found the next theorem.


Theorem!
The sum of the angles of a polygon with n sides has

(n – 2).180°

Proof!
Let. P1, P2 , … ,Pn be the vertices of the polygon as shown in the figure. The segments

decompose the polygon into (n – 2) triangles. Since the sum of the angles of a triangle has 180o, it follows that the sum of the angles of the polygon has (n – 2).180°.
4.10 Condition of th. 4.1

That's it some basic properties of polygon. In next sub chapter, we will learn more about the properties of polygon.

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