# The Modern GMAT’s Ancient Roots

Posted by on Sep 20th 2013

Mathematics credits its most famous formula to a legendary Greek, Pythagoras of Samos.

This gem of trigonometry has had far-reaching impact in every sphere of mathematics and appears quite frequently on the GMAT.

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).

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).

Memorize these common Pythagorean triples.

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. The most common example is the 3-4-5 right triangle.

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.

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.

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