The length of the hypotenuse of a right triangle can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.

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1. If you are given the length of the shortest leg (opposite the 30-degree angle,) simply multiply the leg length by 2 to find the length of the hypotenuse.
2. If you are given the length of the longer leg (opposite the 60-degree angle,) multiply that length by 2/Sqrt(3) to find the length of the hypotenuse.

30-60-90 Triangle Theorem

1. The hypotenuse (the triangle’s longest side) is always twice the length of the short leg.
2. The length of the longer leg is the short leg’s length times √3.
3. If you know the length of any one side of a 30-60-90 triangle, you can find the missing side lengths.

A 30-60-90 triangle is a special right triangle whose angles are 30º, 60º, and 90º. The triangle is special because its side lengths are always in the ratio of 1: √3:2. Any triangle of the form 30-60-90 can be solved without applying long-step methods such as the Pythagorean Theorem and trigonometric functions.

Tips for Remembering the 30-60-90 Rules Remembering the 30-60-90 triangle rules is a matter of remembering the ratio of 1: √3 : 2, and knowing that the shortest side length is always opposite the shortest angle (30°) and the longest side length is always opposite the largest angle (90°).

What is a Triangle? A triangle is a right triangle with angle measures of 30º, 60º, and 90º (the right angle). Because the angles are always in that ratio, the sides are also always in the same ratio to each other.

This means that the ratio of the lengths of the shortest side to the hypotenuse of any right triangle is 1:2. Therefore, If a triangle is a right triangle, the ratio of the sides (short leg:long leg:hypotenuse) is 1:√3:2.

The Pythagorean theorem states that a2 + b2 = c2 in a right triangle where c is the longest side. You can use this equation to figure out the length of one side if you have the lengths of the other two. The figure shows two right triangles that are each missing one side’s measure.

This means we are given two sides and one angle that is not the included angle. In this case, use The Law of Sines first to find either one of the other two angles, then use Angles of a Triangle to find the third angle, then The Law of Sines again to find the final side.

A 3-4-5 right triangle is a triangle whose side lengths are in the ratio of 3:4:5. In other words, a 3-4-5 triangle has the ratio of the sides in whole numbers called Pythagorean Triples.

“SOHCAHTOA” is a helpful mnemonic for remembering the definitions of the trigonometric functions sine, cosine, and tangent i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent, (1) (2) (3) Other mnemonics include.