In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
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06/06/2021 12:23 pm
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In an equilateral triangle, prove that three times the square of one side is equal to four times the square of one of its altitudes.
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06/06/2021 12:24 pm
Let the sides of the equilateral triangle be of length a, and AE be the altitude of ΔABC.
∴ BE = EC = BC/2 = a/2
In ΔABE, by Pythagoras Theorem, we get
AB^{2} = AE^{2} + BE^{2}
\(a^2 = AE^2 + (\frac{a}{2})^2\)
\(AE^2 = a^2  \frac{a^2}{4}\)
\(AE^2 = \frac{3a^2}{4}\)
4AE^{2} = 3a^{2}
⇒ 4 × (Square of altitude) = 3 × (Square of one side)
Hence, proved.
This post was modified 3 years ago by Raavi Tiwari
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