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Serious Geometry Help Needed: Ratio of Sides for a 30-60-90 Triangle?

Hey, pals. As if high school math wasn't enough, I'm prepping for the Math section of the ACT and stumbled upon this: ratio of sides for a 30-60-90 triangle. Anyone know what this ratio is and how can I remember it for the test?

15 days ago

Absolutely! The ratio of the sides of a 30-60-90 triangle is a pretty straightforward relationship that you'll often see on standardized tests like the ACT. In a 30-60-90 triangle, the lengths of the sides are as follows:

- The side opposite the 30° angle, also known as the shortest side, is often represented as x (or sometimes "a" in textbooks).

- The side opposite the 60° angle, which is the longer leg, is equal to x√3 (or "a√3").

- The side opposite the 90° angle, also known as the hypotenuse, is 2x (or 2a).

So, the ratio is 1:√3:2 (or sometimes written as x : x√3 : 2x).

To remember it, there's a little trick that I find useful. Think about the angles as time: "half past" (30 minutes past the hour, which represents our 30° angle), an "hour and a half" (or 90 minutes, standing for our 90° angle). Then picture what's left half way in between: "a full hour", which is 60 minutes for the 60° angle.

The "half past" is the shortest period of time, so it represents the shortest side x. The "hour and a half" is the longest time span, so it stands for the longest side 2x. And "full hour" or the 60 minutes that are in between, correspond to the intermediate side x√3. You might find this or other mnemonics online helpful for recalling this important rule!

15 days ago

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