InterMath | Workshop Support | Write Up Template
Title
Transforming Triangle into Rectangle
Problem Statement
The equilateral triangle with side length e is cut in half and reassembled as the rectangle shown. What is the perimeter of the rectangle?
Problem setup
I am trying to determine the relationship between the sides of an equilateral triangle and the perimeter of a rectangle, which was created from the original equilateral triangle.
Plans to Solve/Investigate the Problem
Predicition:3e because, visually, it looks like 3 sides of the original triangle. E represents the length of the sides in the triangle.
1.
Create the triangle
2.
Find the perimeter of the triangle
3.
Flip the triangle across one side and rotate
4.
Measure the perimeter of the rectangle that is created from the transformation
5.
Make observations about the perimeter of the equilateral triangle in relationship with the perimeter of the rectangle
Investigation/Exploration of the Problem .
First I plan to construct the figure above in GSP.
1.
Create an equilateral triangle. To do this, first I constructed two points and connected them by creating a segment. Then, I rotated the segment 60º and then created a point at the end of the new segment. Then I connected the two points together with a connecting segment.

A j
C
B
2.
Measure the angles and each side length to make sure the figure created is an equilateral triangle. m

ABC = 60.00
 m

ACB = 60.00
 m

CBA = 60.00
 j = 6.23 cm j ' = 6.23 cm m BC = 6.23 cm perimeter

ABG: j +j'+m BC = 18.70 cm
From this we see that the figure created is an equilateral triangle since all the sides are equal and each angle measures 60º.
3.
Construct a midpoint on segment BC to divide the equilateral triangle into two congruent, right triangles.

A j j'
C
G
B
After creating the two new right triangles, it is clear that a rectangle can be formed from the triangles by rotating.
4.
During this step, I tried many times to rotate triangle ABG so it would connect with triangle AGC to form a rectangle. However, I was unsuccessful in my attempts, which caused me to have to approach the problem in a different way.
5.
Now I had to think of a new approach. I decided to try and construct a perpendicular line to segment GC through point C.
A
G j'
C
6.
After I did that, I then created a perpendicular line to line C through point A. Then I created a point where the two lines intersected (point H).

A H
G j'
C
7.
This allowed me to create the rectangle from the two right triangles. I measured both triangles and found that they were both 30º, 60º, 90º triangles. m

CAH = 60.00
 m

AHC = 90.00
 m

HCA = 30.00
 m

GCA = 60.00
 m

AGC = 90.00
 m

GAC = 30.00

8.
Next I measured the lengths of the sides of the rectangle to find the perimeter of the rectangle.
A H m CH = 5.40 cm m GA = 5.40 cm m AH = 3.12 cm m GC = 3.12 cm j'
C
G perimeter of rectang le AHCG m GA+m GC+m CH+m AH = 17.03 cm
9.
Since segment AH and segment GC both measured 3.12, I found that those two sides added together equal 6.23, which is the original length of one side of the equilateral triangle.
10.
Just to make sure that segments AH and GC were actually half of e (original side of triangle), I created a midpoint l on line j’ and measured the new distances created by the two segments. Also, GC was constructed to be this way since G was constructed as the midpoint of BC.

A H
G
I j'
C m AH = 3.12 cm m GC = 3.12 cm m IA = 3.12 cm m GC+m AH = 6.23 cm m IC = 3.12 cm
11.
Then I found that the two remaining, longer sides are the actual height of the equilateral triangle.
12.
To find the height of the equilateral triangle, I used the pythagorean theorem (which will be the sides of the rectangle). So, a²+ (½e)² = e². Then, I need to solve for a². This means a= √¾e which is the long length of the rectangle.
13.
Therefore, this leads me to the conclusion that the perimeter of the rectangle is equal to
(½e + ½e + √¾e + √¾e), which is also ((2*√¾e) + e).
14.
Once I found the perimeter of the rectangle, I observed that the perimeter of the rectangle
(17.03cm) was less than the perimeter of the triangle (18.7cm).
This leads me to the conclusion that the perimeter of the rectangle will be ((2*√¾e) + e), which is one side of the original equilateral triangle plus two of the equilateral triangles’ heights.

Extensions of the Problem
Is it possible to transform a parallelogram into a rectangle?
1.
First I will create a parallelogram in GSP.
B
C
A
D
2.
Then I will create a perpendicular line through point B to line segment AD.
C
A
B
D
3.
Now I will take the triangle formed after cutting the parallelogram with the perpendicular and try to make it into a rectangle. Here I translated the triangle ABD, which created a rectangle .
B
D F
A

Author & Contact
Lauren Johnson, Middle Grades Cohort
Lauren_johnson@ecats.gcsu.ed
Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2