EMAT 6690
~
Centroid ~ Incenter ~
For the next two days we will
explore points of concurrency. The point of concurrency
is the point where concurrent lines intersect. This is a
single point.
Materials needed for this
lesson are wax paper, straightedge, and pencil.
Students will need to have
access to Geometer's Sketchpad.
A triangle has many special lines:
medians, angle bisectors, perpendicular bisectors, and altitudes
to name a few. Each triangle has three of the previously
mentioned lines.
A median is a line
drawn from a vertex to the midpoint of the opposite side.
The point of concurrency of the medians is called the centroid
of the triangle.
Investigations that can be used as
teaching tools for the centroid.
Investigation #1
- Step 1
On a sheet of paper or wax paper, draw a large scalene acute
triangle. Make the triangle as large as you can.
- Step 2
Fold to construct the three medians of each side of the triangle.
*Bring the endpoints together and pinch to locate the midpoints,
then construct the medians with your straightedge.
- Step 3
Repeat steps 1 and 2 for an obtuse scalene triangle.
Investigation #2
- Step 1
Take the scalene acute triangle you constructed
in the previous investigation and label it ABC, as shown
below. Do not worry if your triangle is a different shape.
- Step 2
Label the three medians AF, BD, and CE.
Label the centroid G.
- Step 3
Use your compass to answer the following questions: Is
the centroid the same distance from the vertices? From
the three sides? Take a look at how the centroid divides
each median into two parts. Is the centroid the midpoint
of each median?
- Step 4 You should have answered
no to each question in Step 3. Now compare the small part
on each median to the big part on that median. With you
compass or another piece of wax paper, measure GF.
Compare GF with AG by seeing how many times you
can mark off the distance GF on AG. (If,
for example, you can mark off GF three times on AG,
then AG is three times as long as GF.) How
many times as long as GF is AG?
- Step 5 Compare the lengths
of the parts of the other two medians. (Segment BG
is how many times as long as GD? Segment CG
is how many times as long as GE?)
Students should be able to complete
the follow conjecture after completing the student activity with
Geometer's Sketchpad.
The centroid of a triangle
divides each median into two parts so that the distance from the
centroid to the vertex is -(twice)- the distance from the centroid to
the midpoint.
An angle bisector
is a ray that divides an angle into two congruent angles.
The point of concurrency of the angle bisectors is called the
incenter of the triangle.
Investigations that can be used as
teaching tools for the incenter.
Investigation #1
- Step 1
On a sheet of paper or wax paper, draw a large scalene acute
triangle. Make the triangle as large as you can.
- Step 2
Fold to construct the three angle bisectors in each angle of
the triangle. *At each the vertex of the triangle pinch
to locate the position of the angle bisector, then construct
the angle bisectors with your straightedge.
- Step 3
Repeat steps 1 and 2 for an obtuse scalene triangle.
Investigation #2
- Step 1
Take the scalene acute triangle you constructed
in the previous investigation and label it ABC, as shown
below. Do not worry if your triangle is a different shape.
- Step 2
Label the three angle bisectors AF, BD, and CE.
Label the incenter I.
- Step 3
Use your compass to answer the following questions: Is
the incenter the same distance from the vertices? From
the three sides? Take a look at how the incenter divides
each bisector into two parts. Is the incenter the midpoint
of each bisector?
- Step 4
The incenter is not the same distance from the vertices, but
it is the same distance from the three sides. The incenter
is not the midpoint of each bisector.
Student Activity
Return