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Equilateral triangle vs 30 60 90 triangle?

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Equilateral triangle vs 30 60 90 triangle?

by Gmatism » Wed Aug 24, 2011 10:25 am
Hello,

I am trying to figure out why do I use this formula to calculate third side in an 30 60 90 triangle: x(sqrt)3, while to calculate the height in equilateral triangle I have to divide that by two: (x(sqrt)/2.

to calculate height in equilateral triangle you can split it in half and treat it as 30 60 90 triangle. Sides in 30 60 90 triangle are x, 2x and x(sqrt)3. this last side is the same as height in equilateral triangle.

But the formula for height in eqilateral triangle is (x(sqrt)/2.
I must be doing something wrong because both side are essentially the same thing.


Why is there a difference between the two when it is the same thing. What am I missing?

Thank you so much!
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by arashyazdiha » Wed Aug 24, 2011 12:21 pm
But the formula for height in eqilateral triangle is (x(sqrt)/2.
It's not true.
Let's forget about the degrees.
consider all the sides in an equilateral triangle is X.
draw the height for this triangle. One side is now divided into half which each is X/2
as you know in right triangles if hypotenuse is C and two other sides are A and B then the following formula holds.
C^2 = A^2 + B^2
So in the above right triangle we have:
(Height)^2 + (X/2)^2 = X^2
So (Height)^2 = 3(X^2)/4 then we can say that Height is (X/2)*sqrt(3)

For more explanations on degrees, you should know that in Right triangles the side in front of the 30 degree angle is half of the hypotenuse and the side in front of the 60 degree angle is sqrt(3)/2 of the hypotenuse(calculations using sin and cos)
Bests
Arash
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by gmatboost » Thu Aug 25, 2011 10:12 pm
The short answer to your question is that in one case, you are calling the side of the equilateral triangle x, but in the other case, you are calling it 2x.

If the equilateral triangle has a side of 2x, the height is x*sqrt(3).
If the equilateral triangle has a side of x, the height is x*sqrt(3)/2.

Along these lines, make sure you take this opportunity to memorize the area of an equilateral triangle with side x: (x^2)*sqrt(3)/4.

More on this here: https://blog.gmatboost.com/2011/08/02/me ... -triangle/
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by Gmatism » Fri Aug 26, 2011 7:23 am
Thank you gmatboost! That definitely explains my false reasoning! How could I not see that before....

Thanks a lot!
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