Day 26 – Parallelograms, Pythagorean Theorem

Parallelograms

We have spent a great deal of time studying triangles, which are the simplest example of a shape you can draw using only straight line segments.  These shapes are called polygons.  A polygon is a figure formed by line segments which bound a portion of the plane (this just means that the ends must meet up).  A polygon has sides and angles, and the vertices of the angles are called the vertices of the polygon.

Just as a three-sided polygon is called a triangle, a four-sided polygon is called a quadrilateral (that’s the topic of the current chapter).  There are names for polygons with more sides as well, but we will not be studying them in this class.

A parallelogram is a quadrilateral in which opposite sides are parallel.  This simple fact (opposite sides parallel) allows us to make a surprising number of conclusions about this shape:

Theorem 1:  The opposite sides of a parallelogram are equal.  So are the opposite angles.

The following two videos show why this is true.  In fact, they show the reverse as well — if opposite sides (or angles) are equal, then the figure must be a parallelogram.

Theorem 2:  The successive angles of a parallelogram are supplementary.

Successive angles are one that are “next to each other” in the figure (another way to say it is that successive angles share a side).   Recall that supplementary means that they add up to 180 degrees.

Theorem 3: The diagonals of a parallelogram bisect each other.

The diagonals refer to the lines we can draw connecting the vertices of opposite angles.  This video shows why this fact is true:

Finally, the area of a parallelogram is A=bh, where b is the base of the parallelogram (one of the sides) and h is the height (the distance from the base to the opposite side).

Pythagorean Theorem

Our final topic today is perhaps the most famous theorem in all of mathematics, the Pythagorean Theorem.  The Pythagorean Theorem states that in any right triangle with legs of length a and b, and hypotenuse of length c, we have a^2 + b^2 = c^2.  There are a huge number of different proofs of this theorem — the following one is based on an idea we have studied already in this class, similar triangles:

The Pythagorean Theorem can be used in a great variety of problem types.  The following videos give some examples.

Basic examples (simplifying radicals):

Application problem (mast of a ship):