30-60-90 Triangle Ratio Confusion

[MM]

I’m struggling to know when to multiple by root 3 or divide by root 3 for the ratio of a 30-69-90 triangle (1 : root 3 : 2). Anyone have any advice or tips or ways of memorizing? I got this from ChatGPT, but still not sure

What you know	What you want	Operation
Short leg (30° side) → Long leg (60° side)	Multiply by ( \sqrt{3} )
Long leg (60° side) → Short leg (30° side)	Divide by ( \sqrt{3} )
Short leg (30° side) → Hypotenuse	Multiply by 2
Hypotenuse → Short leg (30° side)	Divide by 2
Long leg (60° side) → Hypotenuse	Multiply by ( \frac{2}{\sqrt{3}} ) or divide by ( \frac{\sqrt{3}}{2} )
Hypotenuse → Long leg (60° side)	Multiply by ( \frac{\sqrt{3}}{2} )

[cylverixxx]

Do you have an example of a confusion you had? It’s not clear what your actual question is.

[MM]

This is a practice question I was working on and I despite knowing the ratio rule for 30-60-90 I got the sides completely wrong

[MM]

Here is another example of where I mislabeled the triangle. How do we know it’s b/2 and not 2b and same for the height?

[cylverixxx]

@MM

Here is another example of where I mislabeled the triangle. How do we know it’s b/2 and not 2b and same for the height?

Do you know that the side opposite to the larger angle must be longer? As such, if the side opposite the 90^{\circ} degree angle is b, then the side opposite the 30^{\circ} must be shorter than that. It can’t possibly be 2b since we know b > 0.

[cylverixxx]

@MM

This is a practice question I was working on and I despite knowing the ratio rule for 30-60-90 I got the sides completely wrong

Once again, start by noting the following relation between side lengths:

AC > AD > DC