Investigations with Geometer’s Sketchpad
Today we’re going to use a program called Geometer’s Sketchpad (GSP) to do some activities in geometry. The first thing you need to do is open the program. Minimize this window and open GSP.
To begin, we need to become familiar with some of the commands in GSP. Do each of the following activities:
Š Construct a point. Label it K.
Š Construct a segment. Label it AB.
Š Construct a line. Label it CD.
Š Construct a line that is parallel to CD and passes through point K.
Š Construct a line that is perpendicular to CD that also passes through point K. Label the intersection of line CD and the perpendicular F.
Š Construct the segment FK and hide the line FK.
Š Move line CD around on the page. What happens? What happens to segment FK?
Now, let’s take what we’ve learned about constructions and do some investigations with area.
Š Open a new sketch and construct a horizontal line AB.
Š Construct point C somewhere above the line. It doesn’t matter where.
Š Highlight both point C and line AB and construct a parallel line. This new line will pass through point C and be parallel to your original line. This is what you should have:
Š Hide points A, B and C.
Š Construct a segment, label it DE, that goes from the lower line to the upper line.
Š Construct another point on the lower line and label it F. Choose point F and segment DE. Construct a parallel line that passes through point F. Label the intersection of this new line and the upper line G.
Š Now we need to cut this line down to a segment. Choose points F and G, construct a segment. Then highlight line FG outside the segment and hide the line.
Š What’s the name of this shape? Let’s construct the interior. To do this, use the point tool to highlight all four points and then use the ‘construct’ menu and choose ‘Quadrilateral Interior’. You can change the color by going to the ‘display’ menu and choose a different color. Your construction should now look like this:
Š Let’s use GSP to measure the area of your quadrilateral. Make sure the interior of your shape is highlighted and then go to the ‘measure’ menu and choose ‘area’. What is the area of your quadrilateral?
Š Play around with your shape. What happens if you move D? E? F? Move line EG up or down. Which of these movements changes the area of your quadrilateral and which don’t? Why do you think this is so?
How can you find the height of your parallelogram?
Ok, now find the height.
Š Now measure the length of segment DF. Use these measurements to find the area of your parallelogram.
Š Write a formula for finding the area of a parallelogram using A for area, b for base and h for height.
Now, let’s use your parallelogram to find the relationship between area of the parallelogram and area of a triangle.
How can you do this?
Before you construct the diagonal, hide the interior of the parallelogram by using the point tool to highlight the interior and choosing ‘hide interior’ from the ‘display’ menu. Once you’ve constructed the diagonal, construct the interior of one of the triangles.
Š Use the measure tool to find the area of your triangle. How is it related to the area of your parallelogram? Drag point E around. What happens to the area? What happens when H and D become the same point? Write a formula for the area of a triangle using A for area, b for base and h for height.
Now, let’s construct a triangle, divide it into regions and see what we can find out about these areas.
Š Construct triangle ABC. Use the segment tool for this construction.
Š Use the ‘midpoint’ tool in the ‘construct’ menu to find the midpoints of sides AB and BC. Label these points D, E. Place point F anywhere on side AC.
Š Construct segments DF and EF. Construct the interiors of all regions, using different colors for the sections.
Move point F along segment AC. How do you think the areas of the triangles are related to the area of the quadrilateral? Be sure to write down your observations when F is the same point as A and C and also when F is the midpoint.
Now measure the area of your triangles and quadrilateral. Were you right? Watch the measurements of all areas as you move F along the segment. Write down your observations.