InterMath | Geometry | Quadrilaterals
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
If you have an equilateral triangle where each side of the triangle is equal to e, what is the
perimeter of a rectangle composed of the equilateral triangle cut in half where the width of the
rectangle would be equal to ½ e and the diagonal of the rectangle is e?
Plans to Solve/Investigate the Problem
First I will draw an equilateral triangle in GSP by using circles. Then I will cut the equilateral
triangle into two equal triangles. One of these triangles I will use as a base for my rectangle.
Using the perpendicular function I will construct the other two sides of the rectangle. After
having drawn the rectangle in GSP I will use the equation for the perimeter of a rectangle to
determine the perimeter of the rectangle knowing what the sides of the triangle are equal to.
Investigation/Exploration of the Problem
1. I draw a circle in GSP and construct a line segment from the center of the circle to a point on
the circle.
2. Using the point on the circle and the center point on the circle I constructed another circle.
3. Where the two circles meet (point B in the below picture) I constructed a point.
4. To construct an equilateral triangle I connected the three points made from the two circles.
Figure 1 is a picture of triangle ABC formed from the two circles:
5. Next I hid the two circles leaving only the equilateral triangle and labeled the sides e.
6. Using the midpoint function I constructed a line that cut the equilateral triangle in half.
Figure 2 is a picture of triangle ABC cut in half
7. Using the perpendicular function I drew a perpendicular line to line segment BD through point
B.
8. Using the perpendicular function I drew a perpendicular line to line segment CD through point
C.
9. I draw a point at the intersection of the two perpendicular lines and label it point F.
Figure 1:
Figure 2:
B
B
e
e
A
C
A
D
C
e
10. By hiding the perpendicular lines and the other half of the triangle (triangle ABD) I have
formed the rectangle out of two halves of an equilateral triangle.
Figure 3 is an example of what the rectangle should look like.
Figure 3:
Figure 4:
F
B
B
e/2
F
e
e
C
D
D
C
e/2
Figure 4 is the rectangle in Figure 3 labeled with the known lengths of its sides
In triangle ABC the sides AB, BC, and AC are all congruent. Since I constructed the midpoint
on line segment AC, line segments AD and CD are equal. That means that both AD and CD are
equal to e/2. Also because line segment BD cut line segment AC into two equal halves, the
angles formed are right angles. Since this relationship happened you can use the Pythagorean
Theorem to find the measurement of the two remaining sides which are equal to the length of the
rectangle.
a = line segment CD or line segment BF = e/2
b = line segment CF or line segment BD
c = line segment BC = e
a2 + b2 = c2
b2 = c2 – a2
b = √c2 – a2
b = √e2 – (e/2)2
Therefore, using the formula for the perimeter of a rectangle we get:
P = 2L + 2W
The length is equal to line segments CF and BD, otherwise known as b. The width is equal to
line segments CD or BF which is equal to e/2.
Therefore,
P = 2(√e2 – e2/4) + 2(e/2)
P = 2√e2 – e2/4 + e
The perimeter of the rectangle is e + 2√e2 – e2/4.
Extensions of the Problem
Is it possible to form a square from an equilateral triangle cut in half?
No, it is not possible to form a square from an equilateral triangle being cut in half. This is
because when the triangle is cut into two equal halves the base of the triangle is cut into half of
what it was originally. Therefore, when the two half triangles are put together to form a
rectangle two of the sides will always be smaller than the other two sides.
Author & Contact
Tabitha Davis
I am a junior in the middle grades cohort concentrating in mathematics and English/language
arts.
tabbydavis240@yahoo.com