Lauren Mofield
Title
Rectangle Squares
Problem Statement:
The rectangle pictured consists of two squares placed side by side. The perimeter of
the rectangle is 57 cm. What is the area of the rectangle in square centimeters?
Problem setup
In this problem, I want to find the area of a rectangle, given the perimeter.
Note: The rectangle is divided in half, forming two squares that have four equal sides.
Plans to Solve/Investigate the Problem
Prediction: The area of the rectangle will be smaller than the perimeter, this I know.
Therefore, I think that the area of the rectangle is….
To begin the problem, I will first construct a rectangle, making sure that all angles equal
90° and that the parallel sides are equal in length. I will then find the midpoint to
separate the rectangle into two equal squares. Constructing these figures in GSP allows
me to measure lengths and construct formulas to determine the perimeter of the rectangle
and the area of the rectangle. The formulas I will use are Perimeter = 2L + 2W and
A= length x width. Knowing the perimeter of the rectangle will allow me to determine
the area of the square. I will make sure that all sides of the squares are equal in
measurement.
Investigation/Exploration of the Problem
1. Using the strait-edge tool, construct a line segment, AB.
2. With line segment AB and both vertexes, A and B, selected, go to Construct,
perpendicular lines.
3. Using the point tool, place a vertex anywhere on the perpendicular line that runs
through vertex A. Label this point C.
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4. Select Line Segment AB and vertex
C. Construct a parallel line. A
rectangle has now been constructed.
A
B
5. Place point D on the perpendicular
line that runs through point B, where
a 90° angle is formed by the parallel
line just constructed.
C
6. Now the rectangle must be split into
two squares. To divide the rectangle
equally, select line segment AB.
Construct a midpoint, label the
midpoint E.
7. Select line segment AB and midpoint
E. Construct a perpendicular line.
The rectangle is now divided into two
squares.
8. Select line segment AE, EB, AC, BD,
and CD and construct line segments.
This must be done individually, not
all together.
9. Place point F at the 90° angle formed by the perpendicular line dividing CD in
half. Make line segment EF a segment.
10. Placing the pointer tool above vertex A, select the entire line running through AC
(this is done with one click). Place the pointer tool above vertex E and select that
line running through EF. Place the pointer tool above vertex B and select that line
running through BD. Select the horizontal line running through line segment
CFD by placing the pointer tool to the left of vertex C and selecting the line.
When all four lines are selected, go to display, hide lines.
A
E
B
C
F
D
11. Measure line AB, CD, AC, and BD. Notice that line AB and CD are the same.
Line segment AC and BD also have the same measurement. Two sets of parallel
sides with the same two sets of measurements confirms that the figure is a
rectangle.
Lauren Mofield
Measurements:
AB = 6.93 cm
CD = 6.93 cm
AC = 2.73 cm
BD = 2.73 cm
2 AB+2AC = 19.31 cm
m AC = 2.73 cm
A problem exists within the measurements. The perimeter of
the figure does not add up to 57 cm. All the requirements were
met: a rectangle was constructed and divided in half to form
two equal squares, right? No, take a look at the next set of
measurements to see that, when measuring each side of the
square, the measurements are not identical.
m AE = 3.47 cm
m EB = 3.47 cm
m FE = 2.73 cm
m CF = 3.47 cm
m DB = 2.73 cm m FD = 3.47 cm
Line segments AC, FE, and DB do not
equal in length line segments AE, EB,
CF, and FD. In order to be a square, all
sides must be equal….
So what to do now?
1. Knowing that the perimeter equals 57 cm., and that all sides of the squares inside
have to be equal, the perimeter of the rectangle should be divided by 6. Why
divide a four sided rectangle by 6? Look at the picture below.
A
E
B
C
F
D
This figure does not include two square inside, but two smaller rectangles, as
explained above. However, this picture can be used for reference. The midpoint, E,
divides the rectangle in half forming four equal line segments: line segment AE, EB, CF,
and FD. This means we have four equal measurements. Line segment AC and BD must
be equal as well, being sides of squares. This is where the number 6 comes from when
57 / 6.
2. Dividing 57 by 6 means that each side of the square is equal to 9.5 square cm.
The sides cannot be anything other than 9.5 square cm. because they will not add
up to 57 cm as the perimeter of the rectangle AND keep the definition of a square,
all sides must be equal.
3.
To construct the figure in GSP, use the straightedge tool to construct a line
segment, AB. Measure the line.
4. Select either vertex, A or B and drag the line until the measurement is 19.00 cm.
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5. Select line segment AB and both vertex points, A and B. Construct perpendicular
lines.
6. Use the point tool to construct a point, C, on the perpendicular line to line AB and
running through vertex A.
7. Select vertex A and C and measure the distance. With only vertex C selected,
drag the vertex until the measurement for the line AC is equal to 9.5cm.
m AB = 19.00 cm
AC = 9.50 cm
A
C
8. Use the select tool to select line segment AB and vertex C. Construct a parallel
line. Label the last vertex point, D, where the last 90° angle is formed.
9. Select vertex A and C, construct a segment. Select vertex A and B, construct a
segment. Select vertex B and D, construct a segment. Select vertex C and D,
construct a segment.
10. Select both perpendicular lines and the parallel line. Go to display, hide lines.
B
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A
B
C
D
Measurements:
AC = 9.50 cm
BD = 9.50 cm
AB = 19.00 cm
CD = 19.00 cm
Looking at the measurements, the figure formed is a rectangle.
The figure has two sets of equal sides. Line segment AC is
parallel to line segment BD. Line segment AB is parallel to
CD. How will this rectangle form two equal squares, and cause
all sides to be equal?
11. The rectangle must be divided in half to form two squares. Select line segment
AB and construct a midpoint, E.
12. Select vertex A and E and measure the distance. Then, select vertex B and E and
measure the distance.
13. Select line segment AB and midpoint E. Construct a perpendicular line. Label
vertex F where the line forms a 90° angle.
14. Select vertex points E and F and construct a segment. Hide the perpendicular
line.
Lauren Mofield
AB = 19.00 cm
A
AE = 9.50 cm
AC = 9.50 cm
E
B
BE = 9.50 cm
EF = 9.50 cm
BD = 9.50 cm
FD = 9.50 cm
CF = 9.50 cm
C
F
D
CD = 19.00 cm
Looking at the measurements, two squares have been created within the rectangle.
To find the area of the rectangle, use the formula: A= L * W
Using the calculator on GSP, select the length of the rectangle, AB, and multiply by the
width, AC.
AB x AC = 180.5
The algebraic equations used for this are:
For the small rectangle (square rather), use the equation L x W
For the larger rectangle, use the equation A = 2(L x W)
To get the area of the smaller rectangle, A = AE x AC, 9.5 x 9.5, and = 90.25 square cm
To get the area of the larger rectangle, A = 2 (AE x AC), 2(9.5 x 9.5), and = 180.5
90.25 x 90.25 = 180.5
Therefore, the area of the larger rectangle is double that of the smaller rectangle.
Author & Contact:
Lauren Mofield
Junior student at Georgia College and State University
Middle Grades Education Major
Darlnlulu4@yahoo.com