Same Angle: Triangle and Square
a triangle, how do you construct a square with the same area using a
dynamic geometry software?
This problem asks to give step
by step process to constructing a square with equal area to a given triangle
using the GSP software.
Solve/Investigate the Problem
I began by taking a piece of
notebook paper and cutting out a square. Then I cut the square along the
diagonal and formed two equal triangles. By translating one of the
triangles across the other, I constructed the triangle with an equal area.
I then made observations about the lengths of each of the sides so that I
could relate this back to a GSP construction.
of the Problem
Begin by constructing a square
(I used the tool already in the program for constructing a square) Measure
the area and record. Connect the opposite vertices to form the diagonal of
Find the midpoints of two
consecutive sides and construct a parallel line to the diagonal. Construct
a line extending from side AB which is parallel to side DC. Where these two
parallel lines meet, mark a point.
Now construct a line
perpendicular to the two parallel lines and mark the point of intersection.
Connect this point to the other two points on the nonparallel line to form
a congruent triangle to the triangles formed by the diagonal of the square.
This triangle has the same area as half the area of the square.
Now construct a line from the
midpoint of side AB and point C. Construct a line along side FG. At the
point that they intersect, mark this point, then connect it to point F and
point E. This forms a congruent triangle to triangle EFG. Now you have an isosceles
triangle with an area equal to the original square.
Extensions of the Problem
What happens if you connect
points I and G and look at that area as it compares to the square?