GeoGebra Tutorial 1 – Quadrilaterals & Midpoints

If this is your first time to use GeoGebra, I suggest that you read Introduction to GeoGebra which contains the discussion about the basics of GeoGebra and parts of the GeoGebra window. The assumption in this tutorial is that you are already familiar with the parts of the GeoGebra window.

In the tutorial below, menu commands, located in the menu bar, are in brown bold text, and submenus are denoted by the > symbol. For example, Options>Labeling> New Points Only means, click the Options menu, choose Labeling from the list, then select New Points Only. The GeoGebra tools are denoted by orange texts. For example, New Point means the new point tool.

Problem: Investigate what happens if you connect the the consecutive midpoints of a quadrilateral.

In this tutorial, we use GeoGebra to explore the properties of the midpoints of a quadrilateral. In doing this tutorial, you will learn how to use the following tools:  Move, Midpoint or Center, Segment between Two Points and Polygon.  If you want to view the output of this tutorial, click here.

1.) Open GeoGebra and select Algebra from the Perspective menu .
newpointpng 2.) To construct the vertices of the quadrilateral, click the New Point tool, and then click four distinct places on the Graphics view.
movepng 3.) If the labels of the points are not displayed, click the Move tool, right  click each point and click Show label from the context menu.
segmentpng 4.) Select the Segment tool and click point A and click point B two distinct points to construct segment AB. To construct BC, with the Segment tool still active, click point B and another location to create segment BC. Now, construct CD and AD to complete the quadrilateral. Your drawing should look like Figure 1.

quad1

Figure 1 – Quadrilateral ABCD

movepng

5.) Click the Move tool and move the vertices of the quadrilateral. What do you observe?
midpointpng 6.) To determine the midpoint of each side of the quadrilateral, choose the Midpoint or Center and then click the segments (not the points) in the following order: AB, BC, CD, and AD

movepng

7.) Select the Move tool, right click the midpoints and click Show label from the context menu. After that step, your drawing should look like Figure 2.

quad2

Figure 2 – Quadrilateral ABCD with Midpoints E, F, G and H.

movepng

8.) Move the vertices of the quadrilateral. What do you observe?
polygonpng 9.) To have a better view, connect the midpoints of the quadrilateral using the Polygon tool. To do this, click the Polygon tool and click the points in the following order: Point E, point F, point G, point H and then point E again to close the polygon.

movepng

10.  Move the vertices of the quadrilateral. What do you observe about the figure?
11.  What conjecture can you make based on your observation?

Updated: 11 January 2016 (GeoGebra 5.0)

Sum of the interior angles of a polygon

We were taught that if we let m be the angle sum (the total measure of the interior angles) and n be the number of vertices (corners)  of a polygon, then m = 180(n-2).  For example, a quadrilateral has 4 vertices, so its angle sum is 180(4-2) = 360 degrees.  Similarly, the angle sum of a hexagon (a polygon with 6 sides) is 180(6-2) = 720 degrees.

But where did this formula come from?  Does this formula work for all polygons?  (Note that in this discussion, when we say polygon, we only refer to convex polygons).

Before we answer these questions, let us first have a brief review of some elementary concepts.

Polygons and Interior Angles

A polygon is a closed figure with finite number of sides. In the figures below, ABCDE is a polygon with 5 sides and (5 vertices).  It is clear that the number of sides of a polygon is always equal to the number of its vertices.

A polygon has interior angles.  In the first figure below, angle B measuring 91 degrees is an interior angle of polygon ABCDE. The angle sum m of ABCDE (not drawn to scale) is given by the equation


m = 126 + 91 + 113 + 102 + 108 = 540 degrees.

twopentagons

In the second figure, if we let a_1, a_2 and a_3 be the measure of the interior angles of triangle ABE, then the angle sum m of triangle ABE is given by the equation m = a_1 + a_2 + a_3.

Angle Sum

To generalize our calculation of  angle sum, we use the fact that the angle sum of a triangle is 180 degrees. Notice that any polygon maybe divided into triangles by drawing diagonals from one vertex to all of the non-adjacent vertices.  In the second figure above, the pentagon was divided into three triangles by drawing diagonals from vertex E to the non-adjacent vertices B and C forming BE and CE.  Now let a_k, b_k and c_k, where k = 1, 2, 3 be measures of the interior angles of the three triangles as shown on the second figure.

 

Calculating the angle sum of pentagon ABCDE we have

pentagonanglesum

Notice that the angle measures in the first line of our equation is just a rearrangement of the measures of the interior angles of the three triangles. Hence, the angle sum of the pentagon is equal to the angle sum of the three triangles. Therefore, we can conclude that the sum of the interior angles of a polygon is equal to the angle sum of the number of triangles that can be formed by dividing it using the method described above. Using this conclusion, we will now relate the number of sides of a polygon, the number of triangles that can be formed by drawing diagonals and the polygon’s angle sum.

table

From the table above, we observe that the number of triangles formed is 2 less than the number of sides of the polygon.  This is true, because n - 2 triangles can be formed by drawing diagonals from one of the vertices to n - 3 non-adjacent vertices. Therefore, there the angle sum m of a polygon with n  sides is given by the formula

m = 180(n - 2)

A More Formal Proof

Theorem: The sum of the interior angles of a polygon with n sides is 180(n-2) degrees.

Proof:

Assume a polygon has n sides. Choose an arbitrary vertex, say vertex V.  Then there are n - 3 non-adjacent vertices to vertex V.  If diagonals are drawn from vertex V to all non-adjacent vertices, then n - 2 triangles will be formed.  The sum the interior angles of n -2 triangles is 180(n - 2). Since the angle sum of the polygon with n sides is equal to the sum the interior angles of n - 2 triangles, the angle sum of a polygon with n sides is 180(n-2). \blacksquare

Exercises:

1.)    Find the number the angle sum of a dodecagon (12-sided polygon).

2.)    The angle sum of a polygon is 3240 degrees. What is the number of its sides?

3.)    The measure of one of the angles of a regular polygon is 150. Find its number of sides.

4.)    From this, prove that the sum of the interior angles of a polygon is 360 degrees.

 

Why do we reverse/flip the inequality sign?

You have probably remembered in Algebra that if we multiply an inequality by a negative number, then the inequality sign should be flipped or reversed. For example, if we want to find the solution of the inequality -\frac{1}{2}x > 8, we multiply both sides by -2  and reverse the greater than sign giving us x < -16. Now, why did the > sign became <?

If we generalize the statements above, suppose we have two numbers, say, a and b such that a > b, if we multiply them to a negative number c, instead of having  ac > bc,  the answer should be ac < bc.

Before we proceed with our discussion, let us first remember 2 basic concepts we have learned in elementary mathematics:

  1. The number line is arranged in such a way that the negative numbers are at the left hand side of 0 and the positive numbers are at its right hand side such as shown in Figure 1.
  2. If we have 2 numbers a and b, then  a > b if a is at the right of b on the number line. For example, in Figure 1, 2 > -1 since 2 is at the right of -1.
Number line

Figure 1 – The number line

For specific values, let’s choose a = 2 and b = -1 as shown in the diagram above and choose c = -1. Note that we will  just use these values for discussion purposes, but we may take any values. It would help, if we think of a and b as two points on the number line with a as a blue point on the right b, a red point.

And note that before multiplying with a negative number, VALUE OF BLUE POINT > VALUE OF RED POINT.

Since a and b are variables, we need to multiply all the numbers on the number line by -1. This is to ensure that whatever values we choose for a and b, we multiply them by -1. If we multiply every number on the number line by -1, the geometric consequence would be a number line with negative numbers on the right hand side of 0, and positive numbers at the left hand side of 0 as shown in Figure 2.

num4-1

Figure 2 – Afer multiplying all numbers on the number line by -1

But negative numbers should be at the left hand side of 0 so we reverse its position by rotating it 180 degrees from any point of rotation (for example, 0).  The resulting figure is shown in Figure 3.

Figure 2 - All numbers in the number line were multiplied by -1

Notice that the blue and red points changed order and that the blue point is now at the left of the red point. Therefore, VALUE OF BLUE POINTVALUE OF RED POINT. That is, why the inequality sign was reversed.

Summarizing, multiplying an inequality by a negative number is the same as reversing their order on the number line. That is, if a, b and c are real numbers, a > b and c<0, then ac < bc.

Our summary above is actually a mathematical theorem. The proof of this is shown below. It is a very easy proof, so, I suppose, that you would be able to understand it.

Theorem: If a, b and c are real numbers, with a > b and c<0, then ac < bc.

Proof:

Subtracting b from both sides, we have a - b>0.

Now, a - b>0 means a - b is positive.

Since c is negative, therefore, c(a - b) is negative (negative multiplied by positive is negative)

Since c(a - b) is negative, therefore, c(a - b) < 0.

Distributing c, we have ac - bc < 0.

Adding bc to both sides, we have ac < bc which is what we want to show .\blacksquare

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