Now that you’re familiar with different types of triangles and understand how to compare separate triangles, I’d like to introduce you to your new best friend the Right Triangle. This triangle is special. It contains a 90-degree angle known as the right angle, and it’s one that is going to pop up quite a bit in other courses after you’re finished with Geometry. If you only master one unit this year, let it be this one! While the general triangle theorems and postulates are applicable to all triangles, right triangles exhibit a distinct set of unique properties that become essential knowledge as you progress in both mathematics and physics. Whether you are a new geometry student or an Algebra II/Trigonometry or Physics student in need of a refresher on fundamental concepts like the Pythagorean theorem or trigonometric ratios (SOH-CAH-TOA), you need this now. I don't care whether you download it and print it out or access it online, just get this information in your brain.
The Pythagorean Theorem:
This a cornerstone of right triangle relationships, and says that in a right triangle, the square of the length of the hypoteneuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Another way to say this is that the sum of the squares of the two shorter side lengths is equal to the square of the length of the longest side. This theorem allows you to always be able to find a third unknown side length if you're given two. What’s even crazier is that if you form squares using the side lengths of each triangle, the combined areas of the two smaller triangles will equal the area of the third triangle.
If you end up taking 2-D physics, right triangles play a crucial role in understanding how things move. Newsflash: there is no such thing as things moving in slanted pathways in physics. Huh? What? It’s true. Anything moving at an angle is actually moving forward or backward, and up or down at the same time. The intersection of these two lines of motion forms a right angle and the hypoteneuse is the motion you see with your eyes. This means that the principles of right triangles are key components in analyzing forces, and projectile motion. The ability to discern and apply the properties of right triangles becomes a valuable skill for physics students when predicting the behavior of systems.
SOH-CAH-TOA is a mnemonic used in trigonometry to remember the three primary trigonometric ratios: Sine, Cosine, and Tangent. Each letter represents a specific ratio associated with the angles of a right triangle. Sine (S) is the ratio of the length of the side that is opposite an angle to length of the hypotenuse (the longest side, opposite the right angle). Cosine (C) is the ratio of the length of the adjacent side (the side touching the angle that is not the hypotenuse) to the hypotenuse. Tangent (T) is the ratio of the length of the side opposite an angle to the adjacent side. By memorizing this you can quickly find all unknown parts of a triangle given two known parts if you use the correct ratio.
As you progress to Trigonometry, the relevance of right triangles and their special properties becomes even more pronounced. Trigonometry, a explores the relationships between angles and sides in right triangles, and then uses those relationships to start looking at wave cycles. This means that the trigonometric ratios — sine, cosine, and tangent — encapsulated by the memorable acronyms SOH-CAH-TOA, become indispensable tools. Right now, you’re just learning what they are but later they will form the basis for trigonometric functions, laying the groundwork for more sophisticated mathematical applications.
Get this one nailed down everyone. The exploration of right triangles goes beyond the confines of geometry and cross-pollinates with more complicated mathematics and physics that you're going to run into later.