Time for MATH MINUTE! (provide your favorite theme music
here).
Get out
your paper & pencil, because I have a new puzzle for you!!
Hints/Responses: PUZZLE #86 Squares inside squares
A. In the
picture below there is a 12x12 square ABCD with another square EFGH inside it
which is formed by connecting the midpoints of the sides of ABCD. Repeating
this operation we get a 3rd square IJKL inside the smaller square
EFGH.
What
is the size of the smallest square IJKL?
By
estimation it appears that the smallest square IJKL inside ABCD has sides of
length 6.
The
corner triangles, like AEF, are 45-45-90 triangles, since the corners are
square and the sides are of length 6. So the hypotenuse is of length 6(sqrt2).
This
means the inner triangles, like IFJ, are also 45-45-90 triangles, because they
have a 90 deg. angle (why?) and sides of length 3(sqrt2). This makes the
hypotenuse of length 3(sqrt2)sqrt2 = 6.
So
it is true that the smallest square IJKL inside ABCD has sides of exactly
length 6.
B. If we
repeat the above construction to get a smaller square QRST inside a square MNOP
inside square IJKL, what is the size of this square?
Using the above
technique we know that it will have sides of length half that of square IJKL,
hence they will have length 3.
Is
there a pattern here?
YES! Each time you
build a square inside as above you get sides which are half the length of the
sides of the large square.
If
we were to repeat the process again, what size square would we get? Can you
make a mathematically educated guess?
YES, length is 3/2,
the size is half the length of the side of the large square.
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).