Question

Prove that , 3 times the square of any side of an equilateral triangle is equal to 4 times square of an altitude​

Answer 1

$\it\pink{To \ prove:}$

$\sf{3 \ times \ the \ square \ of \ any \ side \ of \ an}$

$\sf{equilateral \ triangle \ is \ equal \ to \ 4 \ times \ the}$

$\sf{square \ of \ an \ altitude. }$

$\it\green{\underline{\underline{Proof:}}}$

$\sf{Let \ the \ side \ of \ an \ equilateral \ triangle \ be}$

$\sf{a \ units.}$

$\sf{In \ \triangle ADB, \ \angle \ ADB=90°}$

$\sf{By \ 30^\circ-60^\circ-90^\circ \ triangle \ property}$

$\sf{Altitude=\dfrac{\sqrt3}{2}\times \ hypotenuse}$

$\sf{\therefore{Altitude=\dfrac{a\sqrt3}{2}}}$

$\sf{Now,}$

$\sf{3 \ times \ Side^{2}=3a^{2}...(1)}$

$\sf{Also,}$

$\sf{4 \ times \ Altitude^{2}=4\times(\dfrac{a\sqrt3}{2})^{2}}$

$\sf{=4\times\dfrac{3a^{2}}{4}}$

$\sf{=3a^{2}...(2)}$

$\sf{from \ (1) \ and \ (2),}$

$\sf\purple{\tt{3 \ times \ the \ square \ of \ any \ side \ of \ an}}$

$\sf\purple{\tt{equilateral \ triangle \ is \ equal \ to \ 4}}$

$\sf\purple{\tt{times \ the \ square \ of \ an \ altitude. }}$

$\sf{Hence \ proved.}$