An equilateral triangle inscribed in a parabola y^2 =4x. One of its vertices is at the vertex of the parabola. Find the length of each side of the triangle in units.


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Question Number 201660 by LimPorly last updated on 10/Dec/23

$A n e q u i l a t e r a l t r i a n g l e i n s c r i b e d i n a p a r a b o l a$ $y^{2} = 4 x . O n e o f i t s v e r t i c e s i s a t t h e v e r t e x o f t h e p a r a b o l a .$ $F i n d t h e l e n g t h o f e a c h s i d e o f t h e t r i a n g l e i n u n i t s .$
	Answered by som(math1967) last updated on 10/Dec/23

	$s l o p e o f A B = t a n 30 = \frac{1}{\sqrt{3}}$ $e q u n . o f A B y = \frac{1}{\sqrt{3}} x$ $A B a n d p a r a b o l a i n t e r s e c t a t A, B, C$ $y^{2} = 4 x$ $\Rightarrow \frac{x^{2}}{3} = 4 x \Rightarrow x^{2} - 12 x = 0$ $∴ x = 0, 12$ $x = 12 ∴ y = 4 \sqrt{3}, - 4 \sqrt{3}$ $c o o r d i n a t e o f B (12, 4 \sqrt{3})$ $A B = \sqrt{12^{2} + 48^{2}} = 12 \sqrt{5} u n i t$
	Commented by som(math1967) last updated on 10/Dec/23

	Commented by LimPorly last updated on 10/Dec/23

	$T h a n k a l o t s i r$
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