# Pythagorean Theorem

Most people know the Pythagorean theorem:

*a ^{2}
+ b^{2}
= c^{2}*

The longest side is **always**
*c*. It is called the **hypotenuse**. The **sum**
of the **squares**
of the other two sides (*a*
and *b*) always equals the square of the hypotenuse. You can say that a bajillion different ways, but it always means this beautiful little equation: *a ^{2}
+ b^{2}
= c^{2}*.

The 3-4-5 Right Triangle:

If you plug in 3 for *a*
and 4 for *b*, you will **always, always, ALWAYS**
get 5.

*3 ^{2}
+ 4^{2}
= c^{2}*

*9 + 16 = c*

^{2}*25 = c*

^{2}*√ 25 = √ c*

^{2}*5 = c*

The sequence 3-4-5 is called a “Pythagorean triple.” It holds true for all the multiples of 3-4-5. For example, multiplying each side by 2 will give you 6 and 8 for the shorter sides, and the hypotenuse will **always**
be 10. So, 3-4-5 is a triple, and so is 6-8-10. So is 9-12-15 and so on to infinity.

Standardized tests frequently use triples in their questions, so you gain an advantage by memorizing some of the most common ones below:

3-4-5 (and multiples)

5-12-13 ( and multiples)

7-24-25 (and multiples)

11-60-61 (and multiples)

The Isosceles Right Triangle:

Standardized tests also love the **isosceles**
right triangle.

The sides of the isosceles right triangle also have a constant ratio **s – s – s√ 2 **. If the equal sides are each 4, then the hypotenuse will **always**
be **4√ 2
**.

The standardized tests for college entrance give answer choices with the √ 2 and do not ask you to solve √ 2 .

The 30-60-90 Right Triangle:

Standardized tests also love the 30-60-90 right triangle, (30-60-90 being the measures of each of the interior angles). These sides also have a constant ratio of **s – s√ 3
– 2s **

So, if you see a 30-60-90 right triangle with a hypotenuse of 8, then the shortest side must be half of 8 or 4. Then you will know that the other side is 4√ 3

The standardized tests for college entrance give answer choices with the √ 3 and do not ask you to solve √ 3 .