Undoubtedly, many students in algebra, geometry and/or trigonometry may have considered just such a question. Why study right triangles? Especially, when the study gets tough and the work becomes more demanding, perhaps the purpose of all the prospective effort might loom large in a young mind; perhaps in any mind. 

It is safe to say that usually purpose and reason drive human actions, which is why “senseless acts” get so much popular play. For example, husbands, rightly or wrongly, are immediately prime suspects when a wife goes missing. People have expectations that a human story will typically make sense and will not just be a kind of random act of the universe. Although…

So, again, what might possibly motivate the study of (right) triangles? 

A polygon is a many-straight-sided, closed figure that you could draw on this page. The simplest of all polygons is a triangle. A right triangle is special in that it contains a 90-degree (or right) angle. One of the simplest of all such right triangles is the famous 3,4,5. That number-triple satisfies the famous Pythagorean equation (or theorem) in our title: namely, 32 + 42= 52 or 9+16 =25 and means that a triangle with sides of 3, 4, 5 contains something notably “precious”: a right angle. 

Such an angle occurs virtually never in nature, but is ubiquitous and often in demand in the work of humans.

Long before Pythagoras, at least since the early Bronze Age, humans have known about the specific 3, 4, 5 right triangle relationship and importantly, that it was a way to establish the vertical from the horizontal. In most early handwork construction, the horizontal may not have been the greatest challenge, but the vertical was and remains vitally important, especially if you do not want the standing walls of a structure to unceremoniously lean and, alas, ultimately fall. 

One method uses a horizontal 4-unit stick (the base) and a movable 5-unit stick (the hypotenuse) connected to a lower end (vertex) of the 4 stick. One merely raises a third or 3-unit stick (altitude) until the unattached end of the 5 stick touches the tip (vertex) of the 3 stick. Once the right triangle is thereby constructed, the 3-unit stick will and must then stand vertically (or nearly so).

Picture early humans struggling for food and making shelter to survive. Subsequently, over millennia, the curiosity and development that results from such human struggle drives a transformation into more and deeper questions about geometry, about angles and about the symbolism and abstraction that led us to what is the current reconstruction of knowledge, printed and summarized, in our contemporary math and science school textbooks. 

The human struggle to solve problems from which much knowledge sprang is very nearly invisible from the perspective of school students or other learners, but the codified conclusions of those struggles are what are often presented.

Is it any wonder that learners ask “Why study this”? Shouldn’t they ask? Shouldn’t educators offer some answers? I am sure some do, but, are there other, perhaps, more transparent, and active ways to engage our young learners in scientific and mathematical knowledge? Because without such knowledge, they face limited future possibilities.

John Pace is a retired math professor. He lives in Honesdale, PA.