Q. 254.5( 21 Votes )

Prove that three times the square of any side of an equilateral triangle is equal to four times the square of the altitude.

Answer :

Given: In ΔABC,


AB = AC = BC


To prove: 3 AB2 = 4 AD2


Theorem Used:


1) If corresponding side and two angles of two triangles are equal the triangles are said to have SAS congruency.


2) Pythagoras theorem:


In a right-angled triangle, the squares of the hypotenuse is equal to the sum of the squares of the other two sides.


Proof:



Let AD BC


In ΔADC and ΔADB


AB = AC (given)


B = C (each 60°)


ADB = ADC (each 90°)


ΔADC ΔADB


BD = DC



As ΔADB is a right triangle right angled at D.


By Pythagoras theorem,


AB2 = AD2 + BD2


AB2 = AD2 + (1/2 BC)2


AB2 = AD2 + (1/4 BC2)


As BC = AB


AB2 = AD2 + (1/4 AB2)




3 AB2 = 4 AD2


Hence proved.


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