Math, The Way it Should Be (Unit 2.2: Triangles)

Here comes another math lesson!  It's all about triangles, probably one of the most important shapes.  However, if you're running a bit behind or are new to the series, you kind find the introductory post here, the start of the first unit is here, and the start of this unit can be found here.  While most of that information is not super necessary to today's lesson, not knowing it will prove problematic later.

Now, about triangles.  On the face, there's not much to say about them.  They have three angles (hence the name) and have three sides.  They are closed shapes and always convex, meaning if you put a rubber band around them it would always touch the sides (and all angles are always less than \(\pi\) radians).  In fact, the angles always add up to \(\pi\) radians (if you're wondering why I haven't mentioned degrees, go here).  This is all well and good, but why are these shapes so important that they get their own lesson?

It's because a special type of triangle, called the right angle, is super important when it comes to splitting up vectors.  But before we get to that, let's talk about three kinds of triangles.  The first type is acute with all angles less than \(\frac{\pi}{2}\), the second type is obtuse with one angle greater than \(\frac{\pi}{2}\), and the last type is the aforementioned right angle with one angle at exactly \(\frac{\pi}{2}\).  These right triangles have a very special property: the sum of the squares of the two sides containing the right angle is equal to the square of the third side (known as the hypotenuse).  This is known as the Pythagorean Theorem and more formally looks like this:  \(a^2 + b^2 = c^2\)

This is the whole point of the lesson.  I don't know if I have mentioned this before, but geometry isn't my favorite part of mathematics so I'm kind of glossing over it to get to the cool stuff.  But how does this relate to anything that could possibly be cool?  Recall the graph we made in Lesson 1.6 regarding apples and oranges?  If you don't, it looked like this:

This graph shows a fruit vector representing a person who has two apples and three oranges.  The vector is affectionately known as (2,3).  What if we want to know how far away this person is from a person with no fruit at all?  We can't just add the apples and oranges because that adds to five and it's pretty obvious that the shortest path between the two points isn't that long.  But you can see that the apple vector and the orange vector form a right angle with the fruit vector forming the hypotenuse of a right triangle.  We then apply the Pythagorean Theorem:

\(a^2 + b^2 = c^2\)

\(a = 2\)

\(b = 3\)

\(c^2 = 2^2 + 3^2\)

\(c^2 = 4 + 9 = 13\)

\(c = \sqrt{13}\)*

We now know that the length of the fruit vector is equal to \(\sqrt{13}\).  Aren't you glad we learned about irrational numbers?  Other things to know about the Pythagorean Theorem:

• It can be extended to any number of dimensions (even just one!).  All one needs to do to find the length of a vector is to square all of the components and then take the square root.
• It is never described by the term "to Pythagorean" which means "to speculate in the style of Pythagoras".  This is a common misconception among engineering students.
• It only applies when applied correctly.  The square of the two smaller sides equals the square of the third side.  It doesn't work in any other combination.  It also only works for right triangles.
• This is a very important thing to remember when we get to trigonometric functions.  Keep it in mind.
• Any three integers that can be arranged as a, b, and c and satisfy this relationship are known as "Pythagorean Triples".  The most common is 3, 4, and 5.  The one I remember best is known as the "Alexander Triple", named for my high school geometry teacher: 20, 21, 29.

This lesson is running long so here's your homework:

• Revisit the lesson on vectors and number lines with this new information in mind.
• Apply the Pythagorean Theorem in one, two, and three dimensions and convince yourself that it's true. If that doesn't do it for you, think of a proof.  This problem can be approached through geometry, algebra, or calculus.

-Lane

*The astute student will notice the lack of the \(\pm\) sign in this problem.  If you did, good.  If you didn't, that's okay but this is a super important point.  You almost always need to include it.  However, in this case we can reject the negative answer because distances are positive and a negative length is what we call a "non-physical" solution.  Math is a powerful tool but sometimes it tells us things that are impossible.  Learning to know when math is speaking nonsense and not listening is an important skill that requires critical thinking and common sense.