Constructing Special Parallelograms

There are three ways to prove a parallelogram is a Rhombus. 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 congruent2. Diagonals are perpendicular to each other3. 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 B

(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

D

C

A B

(5) Connect the four points and label the congruent sides with corresponding tic marks

D

C

A B

(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 )

Rhombus ABDC

AB BD DC CA

There are two ways to prove a parallelogram is a Rectangle. 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 congruent2. One vertex angle is right.

Construction steps for #1 (Diagonals are congruent)

(1) Draw an original diagonal DS

(2) Bisect DS at point P.a) Draw two congruent arcs from each

endpoint to make the “fisheye”b) Connect the corners of the “fisheye”

and label it’s “eyeball”

P

D S

(3) Using half of the diagonal; (span PD or PS) make an image circle equal to ½ DS. Note: make the center point (dot) first.

(4) Then draw two diameters on the circle, and label the endpointsNote: the diameters must pass thru the

center point.

D

B A

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)

Rectangle ACBD

BA bisects CD CD bisects BA parallelogram

andBA CD rectangle

One way to prove a parallelogram is a Square is to following are methods used to make a rhombus with 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 semicirclea) Draw semicircle about Bb) Make interesting arcs from points A

and Xc) Draw line from this intersection thru

Bd) Label the angle right with the “box”

A B X

(4) Label the point on the semicircle perpendicular to B as point 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

D C

A B X

(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

Square ABDC

AB BD DC CA

rhombus andmA = 90⁰ rectangle

Another way to prove a parallelogram is a Square is to following are methods used to make a rectangle 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.c) Draw two congruent arcs from each

endpoint to make the “fisheye”d) Connect the corners of the “fisheye”

and label it’s “eyeball”

P

M N

(3) Using half of the diagonal; (span PD or PS) make an image circle equal to ½ DS. Note: make the center point (dot) first.

(4) Then draw one diameter on the circle, and label the endpoints J and K

D

C

(5) Pull a perpendicular from the center point a) The semicircle is already drawnb) Make interesting arcs from points C

and Dc) Draw line from this intersection thru

the center pointLabel the angle right with the “box”

D

C

(6) Use this line as your second diameter and label it’s endpoints R and S

(7) Connect the four endpoints of the diameters (which now makes them also diagonals)

R D

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)

Square CRDS

BA bisects CD CD bisects BA

andBA CD rectangle

andBA CD rhombus