We can use the Pythagorean theorem to show that the ratio of sides work with the basic 30-60-90 triangle above. Since the right angle is always the largest angle, the hypotenuse is always the longest side using property 2. Not only that, the right angle of a right triangle is always the largest angle-using property 1 again, the other two angles will have to add up to 90º, meaning each of them can’t be more than 90º. This is often how 30-60-90 triangles appear on standardized tests-as a right triangle with an angle measure of 30º or 60º and you are left to figure out that it’s 30-60-90. īased on this information, if a problem says that we have a right triangle and we’re told that one of the angles is 30º, we can use the first property listed to know that the other angle will be 60º. In right triangles, the Pythagorean theorem explains the relationship between the legs and the hypotenuse: the sum of the length of each leg squared equals the length of the hypotenuse squared, or \(a^2+b^2=c^2\).In right triangles, the side opposite the 90º angle is called the hypotenuse, and the other two sides are the legs.In addition, here are a few triangle properties that are specific to right triangles: This means that all 30-60-90 triangles are similar, and we can use this information to solve problems using the similarity. Triangles with the same degree measures are similar and their sides will be in the same ratio to each other.You can see how that applies with to the 30-60-90 triangle above. In any triangle, the side opposite the smallest angle is always the shortest, while the side opposite the largest angle is always the longest.In other words, if you know the measure of two of the angles, you can find the measure of the third by subtracting the measure of the two angles from 180. In any triangle, the angle measures add up to 180º.Here are a few triangle properties to be aware of: How do we know that the side lengths of the 30-60-90 triangle are always in the ratio \(1:\sqrt3:2\) ? While we can use a geometric proof, it’s probably more helpful to review triangle properties, since knowing these properties will help you with other geometry and trigonometry problems. If you recognize the relationship between angles and sides, you won’t have to use triangle properties like the Pythagorean theorem. On standardized tests, this can save you time when solving problems. Knowing this ratio can easily help you identify missing information about a triangle without doing more involved math. Here is an example of a basic 30-60-90 triangle: The side opposite the 90 º angle has the longest length and is equal to \(2x\).The side opposite the 60º angle has a length equal to \(x\sqrt3\).The side opposite the 30º angle is the shortest and the length of it is usually labeled as \(x\).Because the angles are always in that ratio, the sides are also always in the same ratio to each other. Here’s what you need to know about 30-60-90 triangle.Ī 30-60-90 triangle is a right triangle with angle measures of 30 º, 60º, and 90º (the right angle). Because its angles and side ratios are consistent, test makers love to incorporate this triangle into problems, especially on the no-calculator portion of the SAT. The 30-60-90 triangle is a special right triangle, and knowing it can save you a lot of time on standardized tests like the SAT and ACT.