Question

16. An an equilateral triangle, prove that three times the square of one side is equal to four
times the square of one of its altitude s​

Answer 1

ANSWER✔

$\large\underline\bold{GIVEN,}$

$\sf\dashrightarrow \text{ an equilateral triangle,having one altitude .}$

$\sf\therefore \text{ let the side of triangle be x }$

$\sf\therefore QS= SR= \dfrac{QR}{2} = \dfrac{x}{2}\:------\boxed{1}$

$\large\underline\bold{TO\:PROVE,}$

$\sf\dashrightarrow \text{ three times the square of one side is equal to the four times the square of one of its altitude.}$

$\sf\therefore that\:is ,\\ 4(PS)^2=3x^2$

$\large\underline\bold{SOLUTION,}$

$\sf\star IN\: \triangle PRS ,$

$\sf\therefore \text{ by Pythagoras theorem,}$

$\sf\therefore PR^2= PS^2+SR^2$

$\sf\implies (x)^2= PS^2+ \bigg( \dfrac{x}{2} \bigg)^2 \:-----\boxed{from\:1}$

$\sf\implies x^2=PS^2= \dfrac{x^2}{4}$

$\sf\implies -PS^2= \dfrac{x^2}{4} -x^2$

$\sf\implies -PS^2= \dfrac{x^2-4x^2}{4}$

$\sf\implies -4PS^2= -3x^2$

$\sf\implies \cancel{-}\:4PS^2= \cancel{-}\:3x^2$

$\sf\implies 4PS^2=3x^2$

$\large{\boxed{\bf{ \star\:\:4PS^2=3x^2 \:\: \star}}}$

$\large\underline\bold{HENCE,PROVED.}$

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