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Why is the long leg the square root of 3 times bigger then the short leg in a 30 60 90 triangle?
Wiki User
∙ 12y ago
I guess you are asking of the two sides which are not the hypotenuse, why the longer is root 3 times the shorter.
Consider a 30, 60, 90 triangle ABC with ∠A = 30o, ∠B = 90o and ∠C = 60o and so side AC is the hypotenuse and BC is the shortest side.
Reflect the triangle ABC in side BC to create triangle ABE. ∠EAB will be 30o, ∠ABE 90o and ∠BEA 60o.
Side AE = AC.
∠EAC = ∠EAB + ∠BAC
= 30o + 30o
= 60o
So triangle ACE is an equilateral triangle and thus side EC = AC
Since EC = AC and EB = BC,
EC=EB + BC
⇒ AC = BC + BC
⇒ AC = 2BC
Using Pythagoras on triangle ABC to find length AB gives:
AB2 + BC2 = AC2
⇒ AB2 + BC2 = (2BC)2
⇒ AB2 = 4BC2 - BC2
⇒ AB2 = 3BC2
⇒ AB = √3BC
ie the shortest side is root 3 times the other non-hypotenuse side.
Wiki User
∙ 12y ago
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