Time for MATH MINUTE! (provide your favorite theme music
here).
Get out
your paper & pencil, because I have a new puzzle for you!!
PUZZLE #89 Triangles in triangles
A. Make a
triangle ABC and choose a point P in the interior of the triangle.
Now
draw lines through P parallel to the sides of the triangle.
These
3 lines will form 3 triangles with the sides of triangle ABC as follows.
Did
you notice that these 3 triangles look like triangle ABC?
Are
they similar to ABC (have the same angles as ABC)?
Why?
What
about the other white shapes inside ABC. What geometric shapes are they?
Why?
B. If you
make different choices for point P you get different sizes for the 3 triangles,
unless you choose P near the geometric center (centroid*) O of ABC (see figure
below).
At
the centroid point O the 3 triangles appear to be congruent and therefore equal
in area.
Are
they?
Why?
BONUS: The 3 triangles inside ABC change sizes as you
move the point P. Is there a point where the sum of the areas of these 3
triangles is a maximum or a minimum?
Why?
* This is the point where the medians (the lines connecting each
of the vertices with the midpoint of the opposite side) intersect. In physics
its the point where the object has its center of gravity or mass. This point
divides the medians in the ratio of 2 to 1.
Have
fun!
Send
your comments, ideas and solutions before Monday to the email below, and
in the subject line be sure to put
MM in the subject line
Visit
us here online at:
http://www2.potsdam.edu/parksjm/MM1.1.htm
to see the results every Friday.
See you next time on MATH MINUTE! (theme music fades out here).