Same Size Perimeter
Consider the shape shown
below. Can you keep the perimeter the same and make the area smaller?
Larger? If so, draw the new figure.
The challenge presents a
definite shape with some alternative points. We are asked to keep the same
perimeter but attempt to rearrange the construction of the line segments so
that the area is reduced.
Solve/Investigate the Problem
I will begin by constructing
the given figure in GeoSketchPad. I will include
all of the points, so that I may be able to rearrange the line segments
while keeping the integrity of the measurement of those segments. I will
construct the figure by creating a line segment, and then using the Transform
function to rotate the segment to the appropriate position whereby the
figure will be drawn accurately and the measurement of the segment will
always be the same. I will then copy the figure, and rotate selected line
segments to appropriate points, whereby the figure will be changed but the
perimeter will remain the same. I will then measure the areas and compare
to see if the perimeter can remain the same but the area be changed.
of the Problem
I constructed the figures in
GSP as described above. The perimeters, as you can see, did indeed remain
the same, but the areas changed depending on the concavity of the angle.
The more angles that can be concave in, the smaller the area. See diagram
Extensions of the Problem
A possible extension of this
problem would be to continue drawing figures with “n” sides; calculate the
number of different figures that can be calculated, and also calculate the
corresponding areas to see if a pattern develops for the relationship of
the areas in reference to the length and number of sides.
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