In geometry, a parallelogram is a quadrilateral with two sets of parallel sides. The opposite sides of a parallelogram are of equal length, and the opposite angles of a parallelogram are congruent. The three-dimensional counterpart of a parallelogram is a parallelepiped.
Contents |
The properties of having equal opposite sides and opposite angles are shared with the antiparallelogram, a type of non-convex quadrilateral in which the two longer edges cross each other.
Let and let
denote the matrix with columns
and
. Then the area of the parallelogram generated by
and
is equal to
Let and let
. Then the area of the parallelogram generated by
and
is equal to
Let . Then the area of the parallelogram is equivalent to the absolute value of the determinant of a matrix built using a, b and c as rows with the last column padded using ones as follows:
To prove that the diagonals of a parallelogram bisect each other, first note a few pairs of equivalent angles:
Since they are angles that a transversal makes with parallel lines and
.
Also, since they are a pair of vertical angles.
Therefore, since they have the same angles.
From this similarity, we have the ratios
Since , we have
Therefore,
bisects the diagonals
and
.
You can also prove that the diagonals bisect each other, by placing the parallelogram on a coordinate grid, and assign variables to the vertexes, you can show that the diagonals have the same midpoint.
The area formula,
can be derived as follows:
The area of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is
and the area of a single orange triangle is
Therefore, the area of the parallelogram is
An alternative, less mathematically sophisticated method, to show the area is by rearrangement of the perimeter. First, take the two ends of the parallelogram and chop them off to form two more triangles. Each of these two new triangles are equal in every way with the orange triangles. This first step is shown to the right.
The second step is merely swap the left orange triangle with the right blue triangle. Clearly, the two blue triangles plus the blue rectangle have an area equivalent to .
To further demonstrate this, the first image on the right could be printed off and cut up along the lines: