Question

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




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Answer 1

Let ABC be equilateral triangle.

Let AD be perpendicular bisector from A on to BC. So BD = CD = 1/2 BC

ADC is a right angle triangle. So  AC² = AD² + DC²

AC² = AD² + (1/2 AC)²           

    AD² = 3/4 AC²  

     4 AD² = 3 AC²

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Answer 2

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#  FIGURE IS PROVIDED BELOW IN THE ATTACHMENT!

Let ABC be an equilateral triangle.

And let AD Perpendicular to BC

In triangle ADB and Triangle ADC, we have :-

AB = AC ( Given )

∠B = ∠C ( Each equal to 60° )

and ∠ADB = ∠ADC ( equal to 90° )

Therefore, ΔADB ≅ ΔADC

⇒ BD = DC

⇒ BD = DC = 1/2 BC.

ΔADB is a right angle triangle so right angled at D.

∴ AB² = AD² + BD²

AB² = AD² + ( 1/2 BC)²

AB² = AD² + BC²/4

AB² = AD² + AB²/4 ( ∵ BC = AB )

⇒ 3/4 AB² = AD²

⇒ 3AB² = 4 AD²

 

$\huge{\boxed{\sf{HENCE\:PROVED\:!}}}$