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GeoGebra

Constructing Structure

Author:Andrew Baxter

Overview

GeoGebra is not just a drawing tool. When objects (e.g., points) are defined with structure in mind, Geogebra can maintain that structure. Consider the diagram below with segment AB. Click and drag point A (only) and watch what happens to the rest of the objects. You should notice the following:
  • Point A can be moved freely. Point A is called an independent object since its position does not depend on other objects.
  • Moving point A makes Segment 1 change to keep A as its endpoint. Segment 1 depends on point A.
Now consider point B.
  • Point B is also independent and can be moved freely.
  • Moving point B does not change Segment 1.
When you first load the page it looked like Segment 1 connects A to B, but the "drag test" you just performed showed that to be an illusion. Move B away from its original position: dragging A indicates that Segment 1 is actually defined as a segment from point A to the fixed point (3,1) (which is where B started). For contrast, the dashed green segment is defined as the segment from A to B.

Single Segment

Four segments

In the four segments below, you'll find that some points have restricted movement. Drag each point one-by-one to determine which are independent objects and which depend on something (which may be visible or invisible). You can tick/untick the "Show structural elements" checkbox to show/hide the objects which are imposing restraints on the points.

What restrictions apply to the endpoints of Segment 1?

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What restrictions apply to the endpoints of Segment 2?

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What restrictions apply to the endpoints of Segment 3?

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What restrictions apply to the endpoints of Segment 4?

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Four Squares

The applet below shows four squares. Well, they look like squares. Drag vertices to determine the structure that is actually built into each quadrilateral.

Describe the features of the quadrilateral that are preserved when you drag A, B, C, and D.

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Describe the features of the quadrilateral that are preserved when you drag A, B, C, and D.

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Describe the features of the quadrilateral that are preserved when you drag E, F, G, and H.

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Describe the features of the quadrilateral that are preserved when you drag E, F, G, and H.

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Describe the features of the quadrilateral that are preserved when you drag I, J, K, and L.

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Describe the features of the quadrilateral that are preserved when you drag I, J, K, and L.

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Describe the features of the quadrilateral that are preserved when you drag M, N, O, and P.

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Describe the features of the quadrilateral that are preserved when you drag M, N, O, and P.

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Your Turn

Create a new worksheet in GeoGebra. You can use either the classic interface or the Geometry perspective. Use your worksheet to construct the following:
  1. Points A, B, and C which are independent objects.
  2. Draw the segments AB and BC.
  3. Create a line parallel to AB passing through C.
  4. Create a line parallel to BC passing through A.
  5. Mark the intersection of the lines from steps 3 and 4 Call that point D.
  6. Hide the two lines from steps 3 and 4.
  7. Create the polygon (not just the segments) ABCD.
  8. Use the drag test to check that ABCD remains a parallelogram.
Next, construct the midpoint quadrilateral for the parallelogram ABCD.
  1. Mark the midpoints of segments AB, BC, CD, and DA. Call those midpoints E, F, G, H respectively.
  2. Create the polygon EFGH.
  3. Show the measurements for the side-lengths and angles of EFGH.

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