Constructing Special Parallelograms
Rhombus.
There are three ways to prove a parallelogram is a
The following are methods using a
compass and straightedge to construct a parallelogram using one of these methods, based on the original figure
to start from.
1. One pair consecutive sides is congruent
2. Diagonals are perpendicular to each other
3. One diagonal bisects it’s vertex angles
Construction steps for #1 (One pair of consecutive sides is congruent)
(1) Draw an original side AB
(2) Span from A to B, and draw an
arc about point A that passes thru
point B.
(3) Pick a random point on the arc
and label it “C”.
C
A
(4) Using the same span (AB) make
congruent arcs about both point B
and point C such that the new
arcs intercept. Label the
intersection as point D
B
D
C
A
B
D
(5) Connect the four points and label
the congruent sides with
corresponding tic marks
C
A
(6) Make a construction statement in
a box showing that all sides are
congruent. (therefore it’s a
parallelogram opps sides , and
one pair of adjacent sides )
AB  BD  DC  CA
B
 Rhombus
ABDC
Rectangle
There are two ways to prove a parallelogram is a
. The following are methods using a
compass and straightedge to construct a parallelogram using one of these methods, based on the original figure
to start from.
1. Diagonals are congruent
2. One vertex angle is right.
Construction steps for #1 (Diagonals are congruent)
(1) Draw an original diagonal DS
(2) Bisect DS at point P.
Draw two congruent arcs from each
endpoint to make the “fisheye”
b) Connect the corners of the “fisheye”
and label it’s “eyeball”
P
a)
D
(3) Using half of the diagonal; (span
PD or PS) make an image circle
equal to ½ DS.
S
D
B
Note: make the center point (dot) first.
A
(4) Then draw two diameters on the
circle, and label the endpoints
Note: the diameters must pass thru the
center point.
C
(5) Connect the four endpoints of the
diameters (which now makes
them also diagonals)
D
B
A
C
(6) Make a construction statement in
a box showing that the diagonals
bisect each other. (therefore it’s a
parallelogram) and the diagonals
are congruent (therefore it’s a
Rectangle)
BA bisects CD
CD bisects BA
parallelogram
and
BA  CD
rectangle
 Rectangle
ACBD
Square is to following are methods used to make a rhombus with
One way to prove a parallelogram is a
an added feature that makes it also a rectangle.
Construction steps for a square rhombus (one angle is right)
(1) Draw an original side AB
(2) Extend the length past point B to
make it at least twice as long.
A
B
(3) Pull a perpendicular from B using
the span from AB as your
semicircle
a) Draw semicircle about B
b) Make interesting arcs from points A
and X
c) Draw line from this intersection thru
B
d) Label the angle right with the “box”
A
(4) Label the point on the semicircle
perpendicular to B as point C
D
B
X
C
(5) Using the same span (AB) make
congruent arcs about both points
A and C such that the new arcs
intercept and label it “D”.
(6) Connect the four points and label
the congruent sides with
corresponding tic marks
(7) Make a construction statement in
a box showing that all sides are
congruent. (therefore it’s a
rhombus) and one angle is 90
degrees so it’s a rectangle
A
AB  BD  DC  CA
rhombus
and
mA = 90⁰
rectangle
B
X
 Square ABDC
Another way to
Square is to following are methods used to make a rectangle
prove a parallelogram is a
with an added feature that makes it also a rhombus.
Construction steps for a square rectangle (diagonals are perpendicular)
(1) Draw an original diagonal MN
(2) Bisect MN at point P.
P
c)
Draw two congruent arcs from each
endpoint to make the “fisheye”
d) Connect the corners of the “fisheye”
and label it’s “eyeball”
M
N
(3) Using half of the diagonal; (span
PD or PS) make an image circle
equal to ½ DS.
D
Note: make the center point (dot) first.
(4) Then draw one diameter on the
circle, and label the endpoints J
and K
C
(5) Pull a perpendicular from the
center point
D
a) The semicircle is already drawn
b) Make interesting arcs from points C
and D
c) Draw line from this intersection thru
the center point
Label the angle right with the “box”
C
(6) Use this line as your second
diameter and label it’s endpoints
R and S
R
D
(7) Connect the four endpoints of the
diameters (which now makes
them also diagonals)
S
C
(8) Make a construction statement in
a box showing that the diagonals
bisect each other. (therefore it’s a
parallelogram) and the diagonals
are congruent (therefore it’s a
rectangle) and the diagonals are
perpendicular (therefore it’s a
rhombus)
BA bisects CD
CD bisects BA
 Square CRDS
and
BA  CD
rectangle
and
BA  CD
rhombus