Karyn Carson

Investigations with GeometerÕs
Sketchpad

Today
weÕre going to use a program called GeometerÕs 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:

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Construct
a point. Label it K.

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Construct
a segment. Label it AB.

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Construct
a line. Label it CD.

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Construct
a line that is parallel to CD and passes through point K.

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Construct
a line that is perpendicular to CD that also passes through point K. Label the intersection of line CD and
the perpendicular F.

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Construct
the segment FK and hide the line FK.

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Move
line CD around on the page. What
happens? What happens to segment
FK?

Now, letÕs take what weÕve learned about constructions and
do some investigations with area.

__Area__

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Open
a new sketch and construct a horizontal line AB.

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Construct
point C somewhere above the line.
It doesnÕt matter where.

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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:

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Hide
points A, B and C.

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Construct
a segment, label it DE, that goes from the lower line
to the upper line.

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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.

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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.

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WhatÕs
the name of this shape? LetÕs
construct the interior. To do
this, use the point tool to highlight all four points and then use the
ÔconstructÕ menu and choose ÔQuadrilateral InteriorÕ. You can change the color by going to the ÔdisplayÕ menu and
choose a different color. Your
construction should now look like this:

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LetÕs
use GSP to measure the area of your quadrilateral. Make sure the interior of your shape is highlighted and then
go to the ÔmeasureÕ menu and choose ÔareaÕ. What is the area of your quadrilateral?

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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 donÕt? Why
do you think this is so?

How can
you find the height of your parallelogram?

Ok, now
find the height.

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Now
measure the length of segment DF.
Use these measurements to find the area of your parallelogram.

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Write
a formula for finding the area of a parallelogram using A for area, b for base
and h for height.

Now,
letÕs 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 Ôhide interiorÕ from the
ÔdisplayÕ menu. Once youÕve
constructed the diagonal, construct the interior of one of the triangles.

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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,
letÕs construct a triangle, divide it into regions and see what we can find out
about these areas.

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Construct
triangle ABC. Use the segment tool
for this construction.

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Use
the ÔmidpointÕ tool in the ÔconstructÕ menu to find the midpoints of sides AB
and BC. Label these points D,
E. Place point F anywhere on side
AC.

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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.