Use Pythagorean Triples on the SAT Math Test

Every minute counts on the SAT Math test, so going through the whole Pythagorean theorem formula every time you want to find the length of a side in a right triangle is a pain in the posterior (and pocket watch).

• Ratio 3:4:5. In this ratio, if one leg of the triangle is 3, the other leg is 4, and the hypotenuse is 5.

Because this is a ratio, the sides can be in any multiple of these numbers, such as 6:8:10 (two times 3:4:5), 9:12:15 (three times 3:4:5), 27:36:45 (nine times 3:4:5), and so on.

• Ratio

where s stands for the side of the figure. Because two sides are congruent, this formula applies to an isosceles right triangle, also known as a 45-45-90 triangle. If one side is 2, then the other leg is also 2, and the hypotenuse is

Here’s what the triangle looks like:

• Ratio

This ratio is a special formula for the sides of a 30-60-90 triangle.

This type of triangle is a favorite of the test-makers. The important thing to keep in mind here is that the hypotenuse is twice the length of the smallest side, which is opposite the 30-degree angle. If you get a word problem saying, “Given a 30-60-90 triangle of hypotenuse 20, find the area” or “Given a 30-60-90 triangle of hypotenuse 100, find the perimeter,” you can do so because you can find the lengths of the other sides:

Time to stretch those mental triangular muscles. Try this sample problem:

1. In this equilateral triangle, the length of altitude AD is

The answer is Choice (D). Look at the 30-60-90 triangle formed by ABD. The hypotenuse is 12, the original side of the equilateral triangle. The base is 6 because it’s half the hypotenuse. That makes the altitude

according to the ratio.

Remember that the 45-45-90 and 30-60-90 triangle patterns are included in the formula box at the beginning of each Math section, in case you forget them.