Analytic Geometry on GeoGebra - Quadrilaterals

Click on the link for the Analytic Geometry on GeoGebra Applet.  A new window will open with a GeoGebra applet which you will use. 

Part 1:  Constructing the Perfect Parallelogram

§         Select the New Point tool, and make three points on the grid at A (-5, 4), B (4, 6) and C (6, -2).

§         Construct a line segment from A to B, using the line segment tool, and then from B to C.

§         To complete our parallelogram, we need one more point.  This point must be a line parallel to side AB, through point C, and parallel to BC through point A.  Here is how you do it:  Select the parallel line tool.  Then click on line AB and then on point C.  A line should appear through the graph.

§         With the parallel tool still on, click on side CB and then on point A.  A second line should appear through the screen.

§         Select the Intersect Two Object tool and click on the two new lines you created.  A new point D will appear which completes the parallelogram.

§         Select the arrow, and click and drag vertex A, B, or C, and notice that no matter where you drag them, you will always have a perfect parallelogram.  Note that you cannot drag point D because it is dependent on A, B, and C.

Part 2:  Diagonals of a Parallelogram

§         Use the Line Segment tool and construct the two diagonals (Line AC and line BD).

§         Now, construct the midpoint of these two diagonals.  If you only saw one point appear that is good.  The midpoint of both diagonals should be the exact same point.  This is a property of parallelograms.

§         Now click and drag vertices A, B, and C and notice that this property always holds true.

§         To verify this, you can use the Distance or Length tool to measure the lengths of the diagonals, and the midpoints to each vertex.

Geometry Fact:  Diagonals of a parallelogram bisect each other.

Part 3:  The Midpoints of the Sides of Quadrilaterals

§         Use the refresh button to put the applet back to its original form.

§         We are now going to make a quadrilateral.  Select the Polygon tool.  Click on the four points where you want the vertices of your quadrilateral to be.  After the fourth point, click on the first to complete the shape.  You should not see your four sided polygon on the applet.

§         Use the Midpoint tool, and click on each side.

§         Now use the Line Segment tool to join the midpoints of each side to their adjacent side’s midpoint.  This should make a quadrilateral inside the one that you have made.

§         Click on any of the vertices of the original quadrilateral and drag it around.

§         You should notice that no matter what the shape of the exterior quadrilateral you made, the interior quadrilateral will always be a parallelogram.

§         To verify this, select the slope tool and measure the slope of each side of the interior quadrilateral.  You will find that no matter where you drag the vertices of the outside quadrilateral, the slopes of the opposite sides will always be equal.

Geometry Fact:  Joining the midpoints of adjacent sides of any quadrilateral forms a parallelogram.

Part 3:  The Midpoints of a Trapezoid

§         To make a trapezoid, start with three points as with a parallelogram; A, B, and C.

§         Use the Line Segment tool to make sides AB and BC.

§         Use the Parallel line tool to make a line parallel to side AB through point C.

§         On this new line, make a new point anywhere you wish.

§         Join this new point to point A.  Now you have a trapezoid.  You can drag on any vertex and it will always remain a trapezoid.  You can even slide your new point D along the parallel line.

§         Use the Midpoint tool to locate the midpoint of the trapezoid’s two non-parallel sides.

§         Join the two midpoints with the Line Segment tool.

§         Use the slope tool to measure the slope of the two parallel lines, and the third line you just constructed.  The slopes should all be equal.  This means that all three lines are parallel.

§         Use the Length tool to measure the lengths of all three parallel sides.  You should find that the average of the longest and shortest should equal the middle side.  Use a calculator to verify this property.

Geometry Fact:  The line segment joining the midpoints of the non-parallel sides of a trapezoid is parallel to the two parallel sides of the trapezoid, and its length is the average of the lengths of the two parallel sides.

That’s All Folks!