GMAT Tip of the Week: Today’s Date in Geometry History

GMAT Tip of the WeekToday is December 5, or in date form it’s 12/5. And if you hope to score 700+ on the GMAT, you should see those two numbers, 5 and 12, and immediately also think “13”!

Why?

There are certain combinations of numbers that just have to be top of mind when you take the GMAT. The quantitative section goes quickly for almost everyone, and so if you know the following combinations you can save extremely valuable time.

Based on Pythagorean Theorem, a^2 + b^2 = c^2, these four ratios come up frequently with right triangles:

a_______b________c
3_______4________5
5______12_______13
x_______x______x*(sqrt 2)___(in an isosceles right triangle)
x____x*(sqrt3)___2x________(in a 30-60-90 triangle)

These four ratios come up frequently when right triangles are present, so they’re about as high as you can get on the “should I memorize this?” scale. But just as important is using these ratios wisely and appropriately, so make sure that when you see the opportunity for them you keep in mind these two important considerations:

1) These “Pythagorean Triplets” are RATIOS, not just exact numbers.

So a 3-4-5 right triangle could also be a 6-8-10 or 15-20-25, and an isosceles right triangle could very well have dimensions a = 4(sqrt 2), b = 4(sqrt 2), and c = 8 (which would be one of the short sides 4(sqrt 2) multiplied by (sqrt 2) ). An average level question might pair 5 and 12 with you and reward you for quickly seeing 13, while a harder question could make the ratio 15, 36, 39 to reward you for seeing the ratio and not just the exact numbers you memorized.

Similarly, people often memorize the 45-45-90 and 30-60-90 triangles so specifically that the test can completely destroy them by making the “wrong” side carry the radical. If the short sides are 4 and 4, you’ll naturally see the hypotenuse as 4(sqrt 2). But if they were to ask you for the length of the hypotenuse and tell you that the area of the triangle is 4 (so 1/2 * a * b = 4, and with a equal to b you’d have 1/2 a^2 = 4, so a^2 = 8 and the short side then measures 2(sqrt 2)), it’s difficult for many to recognize that the hypotenuse could be an integer. So be careful and know that the above chart gives you *RATIOS* and not fixed numbers or fixed placements for the radical sign that denotes square root.

2) In order to apply these ratios, you MUST know which side is the hypotenuse.

In a classic GMAT trap, they could easily ask you:

What is the perimeter of triangle ABC?

(1) Side AB measures 5 meters.

(2) Side AC measures 12 meters.

And it’s common (in fact a similar problem shows that about 55% of people make this exact mistake) to think “oh well this is a 5-12-13, so both statements together prove that side BC is 13 and I can calculate that the perimeter is 30 meters.” But wait – 5 and 12 only lead to a third side of 13 when you know that 5 and 12 are the short sides. If you don’t know that, the triangle could fit the Pythagorean Theorem with 12 as the hypotenuse, meaning that you’re solving for side b:

5^2 + b^2 = 12^2, so 25 + b^2 = 144, and b then equals the square root of 119.

So while it’s critical that you memorize these four right triangle ratios, it’s just as important that you don’t fall so in love with them that you use them even when they don’t apply.

Important caveats aside, knowing these ratios is crucial for your ability to work quickly on the quant section. For example, a problem that says something like:

In triangle XYZ, side XY, which runs perpendicular to side YZ, measures 24 inches in length. If the longest side of the the triangle is 26 inches, what is the area, in square inches, of triangle XYZ?

(A) 100
(B) 120
(C) 140
(D) 150
(E) 165

Those employing Pythagorean Theorem are in for a fight, calculating a^2 + 24^2 = 26^2, then finding the length of a and calculating the area. But those who know the trusty 5-12-13 triplet can quickly see that if 24 = 12*2 and 26 = 13*2, then the other short side is 5*2 which is 10, and the area then is 1/2 * 10 * 24, which is 120. Knowing these ratios, this is a 30 second problem; without them it could be a slog of over 2 minutes, easily, with a higher degree of difficulty due to the extensive calculations. So on today of all days, Friday, the 5th day of the 12th month, keep that 13th in there as a lucky charm.

On the GMAT, these ratios will get you out of lots of trouble.

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By Brian Galvin