Medians in a Triangle
Name(s):
A median in a triangle connects a vertex with the midpoint of the opposite
side. In previous investigations, you may have discovered properties of
angle bisectors, perpendicular bisectors, and altitudes in a triangle. Would
you care to make a guess about medians? You may see what’s coming, but
there are new things to discover about medians, too.
Sketch and Investigate
1. Construct triangle ABC.
B
2. Construct the midpoints of the
three sides.
3. Construct two of the three medians,
each connecting a vertex with the
midpoint of its opposite side.
If you’ve already
constructed three
medians, select
two of them. Then,
in the Construct
menu, choose
Intersection.
To measure the
distance between
two points, select
them; then, in the
Measure menu,
choose Distance.
Select the two
measurements;
then, in the
Graph menu,
choose Tabulate.
E
G
A
F
C
4. Construct the point of intersection of
the two medians.
5. Construct the third median.
Q1
Using the Text
tool, click once on
the point to show
its label. Doubleclick the label to
change it.
D
What do you notice about this third median? Drag a vertex of the
triangle to confirm that this conjecture holds for any triangle.
6. The point where the medians
intersect is called the centroid.
Show its label and change it
to Ce for centroid.
7. Measure the distance from B to
Ce and the distance from Ce to
the midpoint F.
8. Drag vertices of jABC and
look for a relationship
between BCe and CeF.
9. Make a table with these two
measures.
B
BCe = 2.06 cm
CeF = 1.03 cm
D
E
Ce
A
F
C
BCe
CeF
2.17 cm
1.08 cm
1.00 cm
3.24 cm
0.50 cm
1.62 cm
2.06 cm
1.03 cm
10. Change the triangle and double-click the table values to add
another entry.
11. Keep changing the triangle and adding entries to your table until
you can see a relationship between the distances BCe and CeF.
Exploring Geometry with The Geometer’s Sketchpad
© 2002 Key Curriculum Press
Chapter 3: Triangles • 71
Medians in a Triangle (continued)
Choose Calculate
from the Measure
menu to open the
Calculator. Click
once on a
measurement to
enter it into a
calculation.
12. Based on what you notice about the table entries, use the Calculator to
make an expression with the measures that will remain constant even
as the measures change.
Q2
Write the expression you calculated in step 12.
Q3
Write a conjecture about the way the centroid divides each median in
a triangle.
13. Plot the table data on a piece of graph paper or using the Plot
Points command in the Graph menu. You should get a graph with
several collinear points.
14. Draw or construct a line through any two of the data points and
measure its slope.
Q4
Explain the significance of the slope of the line through
the data points.
Explore More
1. Make a custom tool that constructs the centroid of a triangle. Save
your sketch in the Tool Folder (next to the Sketchpad application
itself on your hard drive) so that the tool will be available for future
investigations of triangle centers.
72 • Chapter 3: Triangles
Exploring Geometry with The Geometer’s Sketchpad
© 2002 Key Curriculum Press
Medians in a Triangle (page 71)
Prerequisites: Students should know, or be
introduced to, the terms midpoint and median. It
would be helpful for students to know how to
classify triangles.
Sketchpad Proficiency: Beginner/Intermediate
Activity Time: 30–40 minutes
Example Sketch: Medians.gsp. This sketch
contains the custom tool Centroid, which constructs
the centroid of a triangle given three points (the
vertices).
Demonstration Sketch: Triangle Points.gsp
Sketch and Investigate
Q1 The third median intersects the other two at
their point of intersection. The three medians
are concurrent.
Q2 Students could record either one of the
calculations below.
BCe
CeF
= 2.000
CeF
BCe
= 0.500
Q3 The centroid divides each median into two
parts. One part is twice the length of the other.
Q4 The slope of the line is 1/2 (or 2, depending on
which way students build their tables). This
represents the ratio between the lengths of the
segments the centroid creates on a median.
Explore More
1.
For tips on making and using custom tools,
see the appropriate sections in the help system.
(Choose Toolbox from the Help menu, then
click on the Custom Tools link.) This custom tool
will be especially useful in the activity The Euler
Segment, later in this chapter. A sample centroid
tool is included in the sketch Triangle Centers.gsp.
232 • Activity Notes—Chapter 3: Triangles
Exploring Geometry with The Geometer’s Sketchpad
© 2002 Key Curriculum Press