GSP Assignment
Centers of a Triangle
This assignment
has a set of activities to help you become familiar with GSP and to review some
basic geometry. Please construct and
investigate each of the following activities for various triangles. Use a text box to write-up a two to three
sentence description of your observations in Activities 1-4. You are asked to observe relationships among
the various centers in Activity 5.
The construction
for each activity should be on a separate page.
Use the “tab” feature of GSP in order to create just one file with different
pages for each construction. To activate
this feature go to FILE, Document Options, Add Page, Blank
Page. You should have one file with 5
pages. Suggested names for the pages
are: centroid,
orthocenter, circumcenter, incenter,
all centers, which appropriately correspond to the 5 activities. Note
that the names of the pages appear on tabs in the bottom of the window,
allowing you to jump from one page to another.
After completing
each investigation you should create a custom tool for the construction. The simplest way to create a custom tool is
to complete the desired construction, select all (either from the menu or by
drawing a box around it), and then using the Custom Tool in the toolbox. To save a custom tool so it is always
available on YOUR computer, it must be saved to the TOOL FOLDER in the folder
containing the Sketchpad application.
1.
The CENTROID (G) of a triangle is the common intersection of the three medians.
A median of a triangle is the segment from a vertex to the midpoint of the
opposite side.
Use
Geometer's Sketchpad (GSP) to construct the centroid
and explore its location for various shapes of triangles.
2. The ORTHOCENTER (H) of a triangle is the common
intersection of the three lines containing the altitudes. An altitude is a
perpendicular segment from a vertex to the line of the opposite side. (Note:
the foot of the perpendicular may be on the extension of the side of the
triangle.) It should be clear that H does not have to be on the segments that
are the altitudes. Rather, H lies on the lines extended along the altitudes.
Use
GSP to construct the orthocenter
H and explore its location for various shapes of triangles. (Make sure your
construction holds for obtuse triangles.)
3. The CIRCUMCENTER (C) of a triangle is the point in the
plane equidistant from the three vertices of the triangle. Since a point
equidistant from two points lies on the perpendicular bisector of the segment
determined by the two points, C is on the perpendicular bisector of each side
of the triangle. Note: C may be outside of the triangle.
Construct
the circumcenter C and explore its location for various
shapes of triangles. It is the center of the CIRCUMCIRCLE
(the circumscribed circle) of the triangle.
4. The INCENTER (I) of a triangle is the point on the
interior of the triangle that is equidistant from the three sides. Since a
point interior to an angle that is equidistant from the two sides of the angle
lies on the angle bisector, then I must be on the angle bisector of each angle
of the triangle.
Use
GSP to find a construction of the incenter
I and explore its location for various shapes of triangles. The incenter is the center of the INCIRCLE (the
inscribed circle) of the triangle.
5. Use GSP to construct G, H, C, and I for the same
triangle. What relationships can you find among G, H, C, and I or subsets of
them? Explore for many shapes of triangles.