InterMath | Workshop Support | Write Up Template
Title
Square in a Square
Problem Statementtatement| Two Squares
Pick a point P on the line segment AB and make squares: A side of one square is AP and a
side of the other square is PB. Where should the point P be located to satisfy the condition
that the sum of the areas of the two squares is a minimum? Explain.
Problem setup
Construct a line segment. Place a point on the line segment at any place and label the point P.
Using point P as the end point from one end of the line segment to point P, create a square.
Using point P as the end point from the other end of the line segment to point P, create another
square. Find the area of each square and find the sum of the area of both squares. If you move
point P along the line segment, what will happen to sum of the areas of the two squares. When
will the sum be at its minimal?
Plans to Solve/Investigate the Problem
You can begin by constructing the line segment and placing an arbitrary point on the line
segment. Construct two squares will the arbitrary point as the end point of the each side of the
square.
Investigation/Exploration of the Problem
You begin by constructing a line segment. Using the point tool, place a point any where on the
line segment. Label your line segment AB. Label your point P. Construct a line segment from
point A to point P. This will be the side of your first square. Using construct and transform, you
will construct your first square. You will then use point P and point B and construct a line
segment. PB will be one side of the other square. Using construct and transform, you will
construct the second square. You have to be sure that you construct the quadrilateral interior of
each square. Find the area of each square. Using the measure tool, calculate the sum of the area
of each square. Now you can drag point P along the line segment and find where the sum of the
areas of the two square would be the smallest. Using this method, you will find that if point P is
the midpoint of the line segment, then the sum of the areas of the two squares would be the
smallest. Looking at the drawing from geometers sketchpad, you can see how this will proof the
point.
MA = 3.43 cm
AJ = 3.43 cm
MB = 3.43 cm
BE = 3.43 cm
Area MAJI = 11.74 cm2
Area MHEB = 11.74 cm2
Perimeter MAJI = 13.71 cm
G F
Perimeter MHEB = 13.71 cm
Area MAJI+ Area MHEB = 23.48 cm2
H E
Area APCD = 4.94 cm2
Area PBFG = 21.44 cm2
Area PBFG+ Area APCD = 26.38 cm2
A P M B
D C
J I
You can also solve this problem using algebraic equations. You can determine that the original
line segment would be one. When you place the arbitrary point P on the line segment, then AP is
X and AB would be 1-X. The area of square APCD would be x 2 . The area of square PBFG
would be 1 x . The formula for the sum of the area of the two squares would be
2
x 2 1 x 2 x 2 2 x 1 . Since this equation gives us the total sum and I want to minimize
2
it, the minimum of the graph of y 2 x 2 2 x 1 would give the x value that minimized the sum
of the areas.
Since the x coordinate of the vertex, which is the minimum point of the parabola, is .5. This
would be the midpoint of the line segment. The location of point P should be at the midpoint to
minimize the sum of the areas of the squares.
Extensions of the Problem
Discuss possible extensions for the problem and explore/investigate at least one of the extensions
you discussed.
Author & Contact
Carole C. Jackson
cjackson@rockdale.k12.ga.us
carolecjackson@yahoo.com
.