Step by step
The question
Question: find the other two sides of a 30-60-90 triangle with a short leg of 6.
1 Half of an equilateral
The angles are 30°, 60° and 90°.
Put a copy under it, upside down.
Together they make a triangle with three equal sides.
That is an equilateral triangle.
All three sides are equal, and every angle is 60°.
The short leg is across from the 30° angle.
It is half of one side.
We call it x.
The other half of that side is x too.
The hypotenuse is one whole side.
So it is 2 times the short leg.
The long leg is the short leg times √3.
2 Find the short leg
The short leg is given: x = 6.
3 Find the hypotenuse
The hypotenuse is 2x: 2 times 6.
So the hypotenuse is 12.
4 Find the long leg
The long leg is x times √3: 6 times √3.
Why √3?
Pythagoras: the hypotenuse squared, 4x², take away x².
That leaves 3x².
Its square root is x times √3.
So the long leg is 6√3.
Careful!
√3 is for the long leg of this triangle.
√2 is for the 45-45-90 triangle.
So the hypotenuse is 12 and the long leg is 6√3.
5 Check
Let's check with Pythagoras: short leg squared plus long leg squared.
The hypotenuse squared: 12² is 144.
Both are 144.
So the sides are right!
Let's look back.
First: half of an equilateral.
Then: find the short leg.
Then: find the hypotenuse.
Then: find the long leg.
Last: check.