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प्रश्न
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.
उत्तर
Let the side of the equilateral triangle be a, and AE be the altitude of ΔABC.
`:. BE = EC = (BC)/2 = a/2`
Applying Pythagoras theorem in ΔABE, we obtain
AB2 = AE2 + BE2
`a^2 = AE^2 + (a/2)^2`
`AE^2 = a^2 - a^2/4`
`AE^2 = (3a^2)/4`
4AE2 = 3a2
⇒ 4 × (Square of altitude) = 3 × (Square of one side)
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