Mathematics credits its most famous formula to a legendary Greek, Pythagoras of Samos. Although Babylonians and Indians had used the formula centuries before his birth, this Greek scholar is widely known as being the first to prove the theorem.
Recently, however, historians have contested whether he was truly the first to discover the formula. Part of the debate arises because Pythagoras and his pupils were part of a secretive school of mathematics which left no written records — and hence no evidence.
Yet regardless of who ultimately discovered the theorem, this gem of trigonometry has had far-reaching impacts in every sphere of mathematics. Surprisingly, it remains simple enough to teach to children in middle-school:
Pythagorean’s theorem applies to any right triangle (a triangle that contains a 90-degree angle). In the equation, a and b represent the two legs (the shorter sides), and c represents the hypotenuse (the longer side, opposite the right angle).
Although every right-triangle must satisfy this equation, a few have the special distinction of having all their sides contain only integers. These special 3-integer combinations are known as Pythagorean triples, and they appear quite frequently on the GMAT. The most common example is the 3-4-5 right triangle.
A new Pythagorean triples can be formed simply by multiplying an existing triple by a constant number. For example, if we multiply the 3-4-5 ratio by 2, we get the new Pythagorean triple, 6-8-10. Multiples of existing ratios can easily be calculated, so the only triples you need to memorize for the exam are these three unique ratios: 3-4-5, 5-12-13, 8-15-17.
Having learned the necessary background, let’s see some applications of this theorem. Consider the following data sufficiency problem:
In the figure below, what is the value of c/a?
(1) a is 25% shorter than b.
(2) b is 20% shorter than c.
To have sufficient data, we must have one and only one possible value for the ratio of c/a. We are given a right triangle, which means that the variables must obey the Pythagorean theorem. Perhaps if we can use the statements to replace the variable b, we might be able to express the equation in terms of only c and a alone. That will probably get us closer to finding the ratio.
If we translate statement 1), we have a = b – 0.25b = 3b/4. Rearranging, we have b=4a/3 . If we substitute this into Pythagorean’s theorem, we have:
Typically, when dealing with the squares of variables, we should consider both negative and positive solutions. In a geometry problem, however, lengths must always be positive numbers, so we know that both a and c can never take on negative values. c/a is therefore simply 5/3 — it is a single value, so we therefore have sufficient data.
We can apply the same technique to statement 2). Translating the statement and rearranging, we get b=4c/5. We substitute this back into Pythagorean’s theorem to find that:
Here, too, c/a = 5/3. Again, we have a single value, so we have sufficient data. Each statement alone provides sufficient data.
Our quick review of the Pythagorean theorem should have helped you get caught up on GMAT trigonometry. Before we finish, try to use what you’ve learned to solve the following challenge problem:
Right triangle ABC is divided into five identical right triangles as shown above. What is the ratio, by length, of AC to BC?