InterMath
Title: Transforming Triangles
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 perimeter of a rectangle that is created from an equilateral triangle. My hypothesis
is that the perimeter of the rectangle will be equal to that of the equilateral triangle because you are constructing
the rectangle from the triangle.
Plans to Solve/Investigate the Problem
Using GSP, I will construct an equilateral triangle. I will measure all sides, angles, and find the perimeter. I
will then rotate one of the sides of the triangle to construct the rectangle. I will measure all sides and find the
perimeter. Then I will compare the perimeter of the equilateral triangle and the rectangle.
Investigation/Exploration of the Problem
I attempted to create a triangle first by triangle to make the sides equal, but this did not work. I could always
get two sides equal, but I had trouble making the third side equal.
m HG = 8.09 cm
m IH = 8.09 cm
IG = 6.20 cm
G
I
H
Then I created a triangle by making a line segment (AB). I highlighted the points and segment. I marked the
second point as the center and rotated it 60 degrees. Then I connected the two points by constructing a segment.
I then measured all sides. They all measured 6 cm. I also measured the angles and they all measured 60
degrees. The angles all measure 60 degrees and the sides all measure 6 cm which makes this an equilateral
triangle. Next, I found the perimeter of the triangle. It measures 18 cm.
BA = 6 cm
AC = 6 cm
CB = 6 cm
mBCA = 60.00
D
mABC = 60.00
A
mCAB = 60.00
B
DA = 3 cm
m CE = 3 cm
Perimeter
CAB = 18 cm
Perimeter AECD = 16 cm
m CD = 5 cm
C
m AE = 5 cm
C
E
Transforming the 2 triangles into a rectangle was a difficult task for me. I first tried to rotate, translate, and
reflect the triangles, but this did not yield success. I then made a perpendicular line to CD. I then constructed a
parallel line through point A. The intersection of these lines formed point E and a successful transformation. I
then found the perimeter of the square and compared it to the perimeter of the triangle. The perimeter of the
equailateral triangle( 18 cm) is larger than the perimeter of the square(16cm). The shorter sides of the rectangle
(CE and DA) are ½ the length of the measurement of the sides of the triangle. CE and DA measure 3 c.m. and
all the sides of the equilateral triangle measure 6 c.m. 3 is exactly half of 6 c.m. Line e is a diagonal when the
rectangle is constructed, whereas it is a side of the equilateral triangle. It is not used to calculate the perimeter.
Note: The preferences for this drawing are to the nearest unit.
Another way to check this is using the formula for finding perimeter.
Perimeter of Triangle ABC = s + s + s
Perimeter of rectangle = 2l + 2 w
Perimeter of Triangle ABC = 6 + 6 + 6
Perimeter of rectangle = 2 (5) + 2 (3)
Perimeter = 18 cm
Perimeter of rectangle = 10 + 6
Perimeter of rectangle = 16 cm
Extensions of the Problem
What is the relationship between the area of the triangle and the area of the rectangle. The triangle and the
rectangle have the same area.
Area of triangle = ½ base x height
Area of triangle = length x width
= ½ (6) (5)
= ½ (30)
=5x3
= 15 cm squared
= 15 cm squared
0.556 = 15
Thi s i s 1/2 base x hei ght which equal s
the area of the tri angl e. 15 c.m.
A
D
B
m CD = 5 cm
AD = 3 cm
E
C
CD i s th e hei ght of the
tri angle .
CD i s al so th e l ength of the
rectangl e
AD i s th e 1/2 the base of
the tri angl e.
AD i s al so th e wi dth of the
rectangl e.
53 = 15
Author & Contact
Chantel Lewis
chantel_lewis@putnam.k12.ga.us
Thi s i s l ength x width whi ch
equal s the a rea of the recta ngle.
15 cm
GPS Connection: Students must understand area, perimeter,
GPS:
M5M1. Students will extend their understanding of area of fundamental geometric plan figures.
d. Find the areas of triangles and parallelograms using formulae.
M5P2. Students will investigate, develop, and evaluate mathematical arguments.
M5P3. Students will use the language of mathematics to express ideas precisely.
Link(s) to resources, references, lesson plans, and/or other materials
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