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How to solve 30-60-90 triangles

Definition of a 30-60-90 triangles, including angles and side lengths

A ???30-60-90??? is a scalene triangle and each side has a different measure.

Since it’s a right triangle, the sides touching the right angle are called the legs of the triangle, it has a long leg and a short leg, and the hypotenuse is the side across from the right angle.

In this diagram, the short leg is ???x???, the long leg is ???x\sqrt{3}???, and the hypotenuse is ???2x???. These are always the ratios in a ???30-60-90??? triangle.

We can use the relationships in the diagram to solve all ???30-60-90??? triangles.

How to solve 30-60-90 triangles


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Solving for the side lengths of a 30-60-90 triangle

Example

If ???x=6???, what is the length of the hypotenuse and the long leg?


The hypotenuse is related to ???x??? by ???2x???. We know ???x=6???, so the hypotenuse is ???2(6)=12??? units. The long leg is related to ???x??? by ???x\sqrt{3}???, so the long leg is ???6\sqrt{3}??? units.


Let’s look at another example.


Example

What are the lengths of sides ???a??? and ???b????

The pattern for the sides of a ???30-60-90??? triangle is ???x??? for the short leg, ???x\sqrt{3}??? for the long leg, and ???2x??? for the hypotenuse. In this case we know the hypotenuse is ???4\sqrt{3}???, so

???2x=4\sqrt{3}???

???x=2\sqrt{3}???

The short leg ???b??? is ???2\sqrt{3}???. The long leg ???a??? is ???x\sqrt{3}???, or

???2\sqrt{3}\sqrt{3}=2(3)=6???

The long leg ???a??? is ???6??? units.


Let’s try another example.


Example

What are the lengths of the missing sides?

We know the long leg is ???5\sqrt{3}???, so

???x\sqrt{3}=5\sqrt{3}???

???x=5??? units

The short leg ???b??? is ???5??? units. Then the hypotenuse is

???2x=2(5)=10??? units

The hypotenuse ???c??? is ???10??? units.


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