Quadrilaterals

By: Valerie Jackson

Kite

Definition: a quadrilateral whose four sides can be grouped into two pairs of congruent sides that are adjacent to each other.

Construction

  1. Create an angle of any sort (<BAC)
  2. Construct a circle with center A radius AB.
  3. Plot points B and C where the circle from #2 intersects the angle from #1.
  4. Construct a circle with center C and a random radius (radius CD.)
  5. Construct a circle with center B and radius CD.
  6. Connect points B and D to create line segment BD.
  7. Connect points C and D to create line segment CD.
Area Formula:

A=(pq)/2

Where p and q are the diagonals


Angle relationships:

The angles between unequal sides are equal


Side relationships:

Two sets of congruent adjacent sides


Diagonal Relationships:

The diagonals intersect at right angles


Symmetries:

A kite has one line of symmetry that goes through the vertexes between congruent sides

Rectangle

Definition: A quadrilateral whose four sides can be grouped into two sets of opposite congruent side and whose all angles are equal to 90 degrees.

Construction

  1. Draw line l and plot two random points (points A and B)
  2. Create a perpendicular line to line l that goes through point B
  3. Create another perpendicular line to line l that goes through point A
  4. Construct a circle with center A radius AC
  5. Construct a circle with center B radius AC
  6. Plot point D where the perpendicular line that goes through B intersects the circle from #5
  7. Connect points C and D to create the line segment CD.
Area Formula:

wl=A

where w is width and l is length


Angle Relationships:

All angles are equal to each other and are equal to 90 degrees


Side Relationships:

Opposite sides are congruent and parallel


Diagonal Relationships:

Diagonals are congruent


Symmetries:

There are two lines of symmetry that both go through the middle of each side and extend to the opposite side.

Square

Definition: A quadrilateral whose four sides are congruent to each other and whose angles are alll equal to 90 Degrees

Construction:

  1. Create a random line segment (Line AB)
  2. Create a perpendicular bisector to line segment AB
  3. Name the bisector line CE
  4. Construct a circle with center C and radius CB
  5. Name the point where the circle and line CE intersect point D
  6. Construct a perpendicular bisector to line CE that goes through point D.
  7. Construct a circle with center D and radius CB and plot point F where the circle intersects with the line segment from #6
  8. Connect points F and B to create line segment FB
Area Formula:

A=a^2

where a is equal to a side length


Angle Relationships:

All angles are equal to each other and are equal to 90 Degrees


Side Relationships:

All side are congruent to each other


Diagonal Relationships:

Diagonals are congruent


Symmetries:

There are four lines of symmetry, two of with are the diagonals and two that go through the midpoint of each line and extend to the midpoint of the opposite side.

Rhombus

Definition: A quadrilateral whose sides all are equal to the same length and whose opposite sides are parallel and whose opposite angles are equal.

Construction

  1. Create a line with two points (Line AB)
  2. Construct a circle with center A radius AB
  3. Create a random point on the circle (point C)
  4. Create a circle with center C radius AB
  5. Create a circle with center B radius AB
  6. Create a point where the circle from #4 and #5 intersect and name that point D
  7. Connect point A to point C to create line segment AC and connect point C to point D to create line segment CD and connect point D to point B to create line segment DB
Area Formula:

A=(pq)/2

where p and q are the diagonals


Angle Relationships:

Opposite angles are equal


Side relationships:

All sides are equal in length and opposite sides are parallel


Diagonal Relationships:

The diagonals intersect at right angles


Symmetries:

There are two symmetries, that both extend through opposite points.

Parallelogram

Definition: A quadrilateral whose opposite sides are equal in length and parallel and where opposite angles are equal.

Construction

  1. Create an angle of any sort (Angle CAB)
  2. Create a circle wit center A and a radius of anything (radius AZ)
  3. Create a circle with center B and radius AZ
  4. Plot point D where the circle from #3 intersects with line AB
  5. Create a circle with center D radius ZY
  6. Plot point E where the circle from #3 and #5 intersect
  7. Construct a circle with center B and radius AC
  8. Plot point F where the circle from #7 intersects the line BE
  9. Create line segment CF by connecting points C and F
Area Formula:

A=bh

where b is base and h is height


Angle Relationships:

opposite angles are congruent and consecutive angle are supplementary


Side Relationships:

Opposite sides are parallel and congruent


Diagonal Relationships:

Each Diagonal cuts the other diagonal in half


Symmetries:

There are no lines of symmetry

Trapozoid

Definition: A quadrilateral that has one pair of parallel sides

Construction

  1. Create an angle of any sort (angle ABD)
  2. Construct a circle with center B radius BA
  3. Create a circle with center D radius BA
  4. Plot point E where the circle from #3 intersects the angle from #1
  5. Create a circle with center E and radius CA
  6. Plot point F where the circles from #3 and #5 intersect
  7. Create a parallel line to line BA that goes through points D and F
  8. Plot the two random points G and H on parallel lines and connect them to create in segment GH
Area Formula:

A={(b1+b2)/2}h

where b1 and b2 are the bases and h is height


Angle relationships:

There are no specific angle relationships


Side relationships:

There is one pair of opposite parallel sides


Diagonal Relationships:

There is no diagonal relationships


Symmetries:

None

Isosceles Trapezoid

Definition: A quadrilateral with one pair of parallel sides and of those sides the sides that are not parallel are equal in length and two angles that touch the same parallel side are equal

Construction:

(Do the same steps to create a trapezoid #1-7)

8. Plot random point H on line BA

9. Construct a circle with radius BD and center H

10. Plot point G where the circle from # 9 intersects with the line DF

11. Connect points G and H to from line segment GH

Area Formula:

A={(b1+b2)/2}h

where b1 and b2 are the bases and h is the height


Angle Relationships:

Angles that touch the same parallel side are equal


Side Relationships:

One set of Parallel sides, and the two sides that are not parallel are the same length


Diagonal Relationships:

Diagonals are congruent


Symmetries:

There is one line of symmetry that goes trough the mid point of the two bases