30 60 90 triangle - how's it useful?

Hi there! I've been seeing a lot about the 30 60 90 triangle and its significance in mathematical problems. Could someone explain the formula for this and how it could help in real SAT or ACT problems? I'm a little confused, so any help would be appreciated.

A 30-60-90 triangle refers to the degree measures of the angles in the triangle. It's a special type of right-angled triangle, where one angle measures 30 degrees, the other 60 degrees, and the right angle is, of course, 90 degrees.

The significance of this triangle in mathematics is that the lengths of its sides form a particular ratio, which can be very beneficial when solving mathematical problems, especially geometry ones on standardized tests. This ratio is: 1 : √3 : 2. More specifically, the side opposite the 30-degree angle measures 1 unit, the side opposite the 60-degree angle measures √3 units, and the hypotenuse (i.e., the side opposite the 90-degree angle) measures 2 units.

For example, if the shortest side (opposite the 30-degree angle) of a 30-60-90 triangle measures 5 units, then you can easily calculate the lengths of the other two sides. The hypotenuse (opposite the 90-degree angle) would be double the shortest side, so it'd be 2 5 = 10 units. The other side (opposite the 60-degree angle) would be √3 times the shortest side, so it would be 5√3 units.

It's important to get comfortable with 30-60-90 triangles because they frequently appear in SAT and ACT problems. Recognizing these triangles and being able to quickly determine side lengths can save time and improve accuracy on the test.

For instance, a SAT question could present you a right triangle with an angle of 30 degrees and the length of the shortest side, and it might ask you to find the length of the hypotenuse or the other side. By quickly recognizing this as a 30-60-90 triangle, you can apply the ratio and solve promptly without resorting to the Pythagorean theorem.