GSP LAB: WORKING WITH THE POINTS OF CONCURRENCY
Part One: Working with the Altitudes
1. Open a new GSP file.
2. Create an acute triangle is GSP using the segment tool (like the one below). Click on the select
tool and open space to de-select everything.
3. Let’s create an altitude by following the instructions here in #3.
Select one of the vertex points (how about the top vertex) and the side opposite that vertex.
Once both the vertex point and the side opposite are highlighted, go to the Construct Menu and
choose Perpendicular Line. You will see a new line that is indeed perpendicular to one of the
sides of our triangle. Click on the select tool and open space to de-select everything. Your
image should look approximately as follows:
Next, use the point tool to plot a point at the location where the perpendicular line intersects
the side.
Next, select the segment tool. Create a segment from the top vertex down to the new point
(over top of the perpendicular line). Then, while the new segment is still highlighted,
immediately go to the Display menu and under Line Width choose Thick. Click on the select tool
and open space to de-select everything. Finally, click on the thinner line (outside of the triangle)
and go to Display and choose Hide Line.
Your altitude is now shown as the thick line segment. See below:
4. Repeat step 3 to create the other two altitudes. At the end, you should see that the three
altitudes intersect at a point (called the orthocenter). Plot a point at the orthocenter and label it
with an O.
O
5.
Now, drag one of the vertices of your triangle to create a right triangle (or at least as close as
you can come to forming a precise right triangle). What do you notice about the location of the
orthocenter?
Answer:
6. Now, drag one of the vertices of your triangle to create an obtuse triangle. The altitude
segments disappear! But, go to the Display menu and then Show All Hidden. The lines are
shown again. When extended, they do indeed cross at a point. We will still call this point the
orthocenter. What do you notice about the location of the orthocenter when we have an
obtuse triangle?
Answer:
PART TWO: FINDING THE LOCATION OF A CIRCUMCENTER
Prior to this part of the lab, recall two definitions:
Circle: The set of all points equidistance from a single point called the center.
Triangle Circumcenter: The point of intersection between the three perpendicular bisectors of a
triangle.
1. Using the segment tool, create an acute triangle ABC as shown.
B
C
A
2.
Follow the instructions here in #2 to construct the three perpendicular bisectors of this triangle.
To do this, first highlight all three sides of the triangle. Then, go to the Construct menu and
choose Midpoint. You will see the midpoints appear.
Next, click on the arrow tool and then open space so that nothing is selected. Click on one of
the midpoints to highlight it. Also, click on the same side of the triangle that your highlighted
midpoint is on. Once both the midpoint and the segment are highlighted, go to the Construct
menu and choose Perpendicular Line. You will see the perpendicular bisector now drawn on
your sketch.
Repeat for the other two sides. When finished with this step, your image should look something
like this:
B
A
C
3. Carefully plot a point at the intersection of the lines. Let’s label the point as CC. This is, of
course, our circumcenter.
4. Using the circle tool, and using the circumcenter as the center, plot a circle that passes through
each of the vertices of the triangle. Make sure to “lock” the control point of the circle onto one
of the vertices of the triangle. Your image should look approximately as follows:
B
CC
A
C
5. Now, drag one of the vertices of your triangle to create a right triangle (or at least as close as
you can come to forming a precise right triangle). What do you notice about the location of the
circumcenter?
Answer:
6. Now, drag one of the vertices of your triangle to create an obtuse triangle. What do you notice
about the location of the circumcenter?
Answer:
GSP LAB: ANSWER SHEET
Part One: Working with the Altitudes
5. Now, drag one of the vertices of your triangle to create a right triangle (or at least as close as
you can come to forming a precise right triangle). What do you notice about the location of the
orthocenter?
Answer:
6. Now, drag one of the vertices of your triangle to create an obtuse triangle. The altitude
segments disappear! But, go to the Display menu and then Show All Hidden. The lines are
shown again. When extended, they do indeed cross at a point. We will still call this point the
orthocenter. What do you notice about the location of the orthocenter when we have an
obtuse triangle?
Answer:
PART TWO: FINDING THE LOCATION OF A CIRCUMCENTER
5. Now, drag one of the vertices of your triangle to create a right triangle (or at least as close as
you can come to forming a precise right triangle). What do you notice about the location of the
circumcenter?
Answer:
6. Now, drag one of the vertices of your triangle to create an obtuse triangle. What do you notice
about the location of the circumcenter?
Answer: