In an equilateral triangle prove that three times the square of one side is equal to four times the square of one of its altitude.
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Answer:
Given:- An equilateral triangle with one of its altitude.
Let a be the side of the equilateral triangle.
∴BE=EC=
2
BC
=
2
a
To prove:- 4AE
2
=3a
2
In △ABE, by pythagoras theorem
AB
2
=AE
2
+BE
2
a
2
=AE
2
+(
2
a
)
2
⇒AE
2
=a
2
−
4
a
2
⇒AE
2
=
4
4a
2
−a
2
⇒AE
2
=
4
3a
2
⇒4AE
2
=3a
2
Hence proved that three times the square of one side is equal to four times the square of one of its altitudes.
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