Step by step
The question
Question: find the other two sides of a 30-60-90 triangle with a long leg of 9√3 cm.
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 long leg is x√3.
So x√3 = 9√3.
Careful!
Half is only for the hypotenuse.
The long leg is x times √3.
Both sides are times √3.
So x = 9 cm.
3 Find the hypotenuse
The hypotenuse is 2x: 2 times 9.
So the hypotenuse is 18 cm.
So the short leg is 9 cm and the hypotenuse is 18 cm.
4 Check
Let's check with Pythagoras: short leg squared plus long leg squared.
The hypotenuse squared: 18² is 324.
Both are 324.
So the sides are right!
Let's look back.
First: half of an equilateral.
Then: find the short leg.
Then: find the hypotenuse.
Last: check.