GMAT Tip of the Week: Epiphany

Happy January 6, or as it is known to many, the day of  the Epiphany…the twelfth day of Christmas.  If your New Year’s Resolution includes getting serious about the GMAT and your  b-school future, epiphanies are a great place to start.

The feast of the Epiphany, in Western Christianity, celebrates primarily the visitation of newborn Jesus Christ by the three kings, who famously bore gifts of gold, frankincense, and myrrh.  And this is a mistake that many GMAT test-takers make when studying – they anticipate “knowledge” as “gifts,” asking questions like “what is the formula?” and “what is the rule?”  But, really, what’s important about the Epiphany is not the gifts themselves, but the revelation (in the Christian tradition of the new Lord to the rest of the world). And for your GMAT study, the revelation/epiphany that comes with newfound (or newly-reviewed) knowledge is exponentially more important than is the knowledge itself.  As you study for the GMAT, allow yourself to have epiphanies and not just “gifts.” 

For example, you might encounter a problem that asks for the area of an equilateral triangle with the length of one side of 2.  And you might, correctly, think “there’s a formula for that!” and look back to your notes to find the formula (Area of an Equilateral Triangle = (s^2 * sqrt3)/4).  You could then plug in 2 for and determine that the area is sqrt 3.  But that process involves your receiving that formula as a “gift,” when what you really want to encounter as you study is an epiphany.  In order to have that epiphany, you need to struggle a little bit – you need to think your way through it, and not just look-up, plug, and chug.  So you should instead, as you study, come up with the thought process on your own, then link that back to the rule later.  Here’s your route to an epiphany with that rule (follow the yonder star…):

* The area of any triangle is 1/2 * base * height.  We’re given one side of the triangle, which will form our base (2), and now need to find the height, which is a perpendicular line between the base and its opposite angle.

* Because an equilateral triangle is perfectly symmetrical (all sides and angles are equal), that perpendicular line will perfectly split the triangle into two 30-60-90 triangles.  Not sure why?  Remember this – the perpendicular angle is, by definition, 90 degrees.  And one of the 60 degree angles (that one that is not split by the height line) will remain 60.  So the remaining angle must be 30.

* If the hypotenuse is 2, the shortest side must be half that (the ratio for 30-60-90 sides is x, x*sqrt 3, 2x).  Or you can intuitively notice that, by splitting that base precisely in half, you’d make the short side of the small triangle half of 2 = 1.

* That all means that the height line is the short side, or half the base, times sqrt 3.  So the height is 1 * sqrt 3.  And the base is 2, so 1/2 * base * height = 1/2 * 2 * sqrt 3 = sqrt 3.

Now for the epiphany – how does that link to the formula?  If an equilateral  triangle has a side of s, then the base is just that same length, s.  And to find the height, you’d cut the base in half (to find the 30-degree side length) and multiply by the square root of 3 (to find the x*sqrt 3 ratio side for the 60-degree side).  So A = 1/2 * b * h gives you, in terms of side length:

1/2 * s * (1/2 * s * sqrt 3)

Combine like terms and you get:

1/2 * 1/2 * s * s * sqrt 3

1/4 * s^2 * sqrt 3

That’s where the formula comes from.  And that “epiphany” (Oh!  That’s why that formula holds) can be crucial for multiple reasons:

1) It’s hard to forget or misremember a formula if you know why it holds – the moment doubt creeps in you can always prove it again

2) Teaching yourself to prove this formula helps you develop the tools to prove to yourself other formulas should the need arise on test day

3) Knowing the 360-degree view of a relationship (not just the snapshot single-form formula) makes you more flexible in applying it, and the GMAT loves to ask you to reverse-engineer the concepts that almost everyone knows forward but that few can piece together backward

On this feast of the epiphany date, make a New Year’s Resolution to seek out epiphanies, and not just memorize formulas, as you study for a 2012 GMAT appointment.  Today, many around the world celebrate the Three Wise Men.  Here’s hoping that, this year, you can become a fourth!

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