Mathematics

An Introduction Quadrilaterals



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Let's get to know our four-sided friends. Sure enough, the quadrilateral is named quite literally. That's "quad" for four, and "lateral" for side. (You didn't think football players made up the word, did you?) But to make sure that everyone is on the same page here, let's give a specific definition to quadrilaterals, shall we?

* A quadrilateral is any closed, two-dimensional figure constructed from four line segments joined in sequence at their endpoints to form four vertices.

If you weren't familiar with some of those words in this context, here are some brief definitions:

* closed the figure does not have any openings, so it has a definite interior and exterior

* two-dimensional flat, like a drawing on a piece of paper

* line segment a piece of a line, with definite endpoints (a line has no end)

* vertices the plural of vertex, it refers to the corners where our line segments connect

There are special types of quadrilaterals, and special properties. Odds are that you know of many, most, or even all of them, even if you don't think about them often. Because they overlap in many cases, there is no clear place to start, but we'll try to cover a good portion of it. I can't include diagrams here, but I'll try to describe things well enough that you can follow along on paper if you wish.

Taking any random quadrilateral you draw or see, there are some qualities you should be aware of. One is that you can always draw two diagonals lines that connect the opposite corners of the quadrilateral. In the most common quadrilaterals those that make a simple four-sided loop, both diagonals are completely inside the quadrilateral. (These are convex quadrilaterals.) You can draw quadrilaterals with the diagonals outside too. For example, a boomerang-shaped figure could have two vertices at the top and bottom, with one line segment from each point extending far out to the right where they eventually meet, and a second line segment from the top and bottom corners that extend only slightly to the right before meeting. In this case, the diagonal from top corner to bottom corner lies entirely outside of the figure, while the other is internal. Even more unusual is the quadrilateral you form by drawing a large "X" and then drawing line segments across the top and bottom. This creates a self-intersecting quadrilateral one in which two of its line segments cross. The diagonals for this figure are both external, with one on the right and one on the left.

Every line segment has a midpoint a point exactly halfway between the two endpoints. The four midpoints of the sides of a quadrilateral can be connected in sequence, forming a new, smaller quadrilateral that has interesting properties. If you try it, you'll see that the opposite sides of the new quadrilateral are parallel. What's more, if you draw the diagonals for the new quadrilateral, they will bisect (cut in half) one another. This makes sense, since they come from the old midpoints, so you can expect that they are passing through the halfway point of the figure and through the halfway points of one another as well. This new and interesting quadrilateral has a specific name. It is called a parallelogram, and can be defined using either of the two properties that we just mentioned.

* A parallelogram is a quadrilateral that has two sets of parallel sides, or

* A parallelogram is a quadrilateral whose diagonals bisect one another

Both definitions are correct, and any quadrilateral having either property will also have the other, making it a parallelogram. Incidentally, try saying parallelogram ten times fast.

Another property of parallelograms is that its opposite sides are equal in length. Also true is that opposite angles are equal. These too can be used as definitions of a parallelogram.

* A parallelogram is a quadrilateral in which all opposing sides are equal in length

* A parallelogram is a quadrilateral in which all opposing angles are equal

Just as parallelograms are a special type of quadrilateral, there are also special types of parallelogram. The rhombus is my personal favorite, and surely must be to any computer fanatic. (It is pronounced ROM-BUS, after all.) A rhombus is a parallelogram whose sides are all of equal length. It has all the properties of a parallelogram, plus its four sides are all equal. It is enough to use the last fact to define a rhombus, however.

* A rhombus is a quadrilateral with four equal sides

Many students protest at this point. "Hey, wait! That's a square." But it isn't, or at least not always. A square is a special case. A rhombus can exist with a variety of angle measures, but a square can only have right angles. (We'll have more on squares soon.) The rhombus is still a parallelogram, so its opposite angles are equal, but not necessarily any specific measure.

We just mentioned "right angles". Since we've never discussed "wrong angles", it obviously means something different. Most students learn that a right angle is a 90 degree angle. That's great, if you're working in a 360 degree system. If you're in radians, then the measure is pi/2. Or you might measure angles in gradients, pie wedges, or percents. The key here is that geometry isn't based on measuring systems, so we'll define a right angle this way instead:

* A right angle is the angle formed when two lines intersect in such a way that four equal angles result

What happens when a parallelogram has a right angle in it? The opposite angle also has to have the same measure. With two right angles in place, half the total angle measure is used up, so the other two have to be right angles as well. Incidentally, we didn't state it before, but the four angles in any convex quadrilateral will add up to 360 degrees (in degrees) or 2 pi (in radians), etc. The result is that one right angle in a parallelogram requires that there be four. When a parallelogram has right angles, it forms a familiar shape, though still not a square. It makes a rectangle.

*A rectangle is a parallelogram with a right angle

It would be possible to define a rectangle in terms of its sides and angles, but since we know the definition of a parallelogram covers that already, it is more efficient to just add on the right angle stipulation, as we did here.

At last, what if we combine the special attributes of rhombuses and rectangles. The resulting figure has four equal sides as well as right angles. We could call it a rectangular rhombus, or a rhomboidal rectangle, but we'll be boring and just call it a square, like everyone else.

* A square is a rhombus with right angles

* A square is a rectangle with four sides of equal length

You don't really need four sides if you know that the angles are all right angles though. If two adjacent angles are equal in length, the right angles assure that the other two will be as well.

*A square is a quadrilateral with four right angles and two adjacent sides equal in length

That doesn't provide much of an advantage, but if you're stuck measuring sides, it can be nice to only have to check two instead of four. There is another shape incidentally, that is a quadrilateral with two adjacent, equal sides. By now, you might be getting tired of quadrilaterals, and feel like telling me to "go fly a kite", and you'd be exactly right.

The kite is a quadrilateral, one in which two adjacent sides are equal in length, and the other two are also equal in length to one another, but the two pair may not be the same as one another. (That would be a rhombus, of course.) In definition form;

* A kite is a quadrilateral with two distinct pair of equal adjacent sides.

The key word "distinct" means that we can tell the two pair apart, so they must be unequal. You might like kites, and not just for flying. They are not parallelograms, and in fact cannot have any parallel sides. Kites have bilateral symmetry, as can be seen in the following properties. Only the shorter of the two diagonals gets bisected in a kite. The longer one is unevenly divided. The angles between sides of unequal length will also be equal, while the angles between sides of equal length will be unequal.

The last distinct quadrilateral (that I can think of) is the trapezoid. I saved this one for last because it is a little obnoxious to define. The problem is that its definition is inconsistent, depending where you look. It has two parallel sides, but some definitions limit it to exactly two, while others leave it open. As a result, you can take your pick, but I'd recommend being consistent when working with them.

* A trapezoid is a quadrilateral with two parallel sides

* A trapezoid is a quadrilateral with exactly two parallel sides

The second definition is the more limiting of the two. The first means that any shape with two parallel sides is also a trapezoid, so that parallelograms (and their subcategories) are also trapezoids. I prefer the more open definition, because parallelograms do follow all the same behaviors as trapezoids, just having their own special traits added in to make them more distinct. Any math performed on a trapezoid also applies to a parallelogram, so it seems unfair to exclude them.

Now, it needs to be mentioned that this is the American definition of trapezoid. In many other countries, the same figure is referred to as a trapezium instead. (This is another reason the shape's definition is obnoxious.) To make things even worse, the opposite name (trapezium in the US, Trapezoid other places) is used to refer to a quadrilateral with no parallel sides. Just know which one fits the local geometrical culture before you go talking parallel lines, and you'll be fine.

Some quadrilaterals can be inscribed on a circle. That's a fancy way of saying that you can draw the quadrilateral so that all four vertices touch the edge of a circle. Such quadrilaterals are called "cyclic quadrilaterals". (Note that's cyclic, not psychic!) For any convex cyclic quadrilateral, the opposing angles will be supplemental to one another that is, they can be added together to form a straight line (180 degrees, pi radians, etc). This makes it very clear that some quadrilaterals cannot be cyclic. In particular, parallelograms cannot be cyclic, with one exception. Because their opposing angles are always equal, one pair adds to more than a line, the other less. The exception? Rectangles (and squares, which are rectangles too) have four matching angles right angles so they are supplemental, and are always cyclic. Some, but not all, kites are cyclic. Only trapezoids that have matching base angles (the angles formed at both ends of one parallel side) can be cyclic. These are known as isosceles trapezoids. Again the rectangle fits into this definition.

That should just about conclude our overview of quadrilaterals. I deliberately avoided the math such as calculating areas and perimeters so that we could focus on geometric properties instead. You can perhaps look for a future article on the calculations for quadrilaterals. Now, to review, let's end with a summary of which quadrilaterals are related.

All squares are rectangles, rhombuses, parallelograms (and trapezoids, by one definition).
All rectangles are parallelograms (and trapezoids, by one definition).
All parallelograms are trapezoids, by one definition.
Kites are a unique category.
Self-intersecting quadrilaterals are a unique category.

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