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Same Angle: Triangle and Square

Problem Statement
Given a triangle, how do you construct a square with the same area using a dynamic geometry software?

Problem setup

This problem asks to give step by step process to constructing a square with equal area to a given triangle using the GSP software.

Plans to 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.

Investigation/Exploration 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 the square.



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?