points (a,a),(-a,-a),and(-root3a,x)

r the vertices of the equilateral triangle. find x.

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points (a,a),(-a,-a),and(-root3a,x)
r the vertices of the equilateral triangle find x

Priyanka Kedia · Accepted Answer

$G i v e n, c o - o r d i n a t e o f A, B a n d C a r e (- \sqrt{3} a, x), (- a, - a) a n d (a, a) S i n c e I n a n e q u i l a t e r a l t r i a n g l e, a l l s i d e s a r e e q u a l . T h e r e f o r e, A B = B C = C A \Rightarrow {A B}^{2} = {B C}^{2} = {C A}^{2} \Rightarrow {A B}^{2} = {B C}^{2} a n d {B C}^{2} = {C A}^{2} W h e n {A B}^{2} = {B C}^{2}, {(- a - (- \sqrt{3} a))}^{2} + {(- a - x)}^{2} = {(a + a)}^{2} + {(a + a)}^{2} (u \sin g d i s \tan c e f o r m u l a, i f A = (x_{1}, y_{1}) a n d B = (x_{2}, y_{2}), t h e n A B = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}) \Rightarrow {(- a + \sqrt{3} a)}^{2} + {(- a - x)}^{2} = {(2 a)}^{2} + {(2 a)}^{2} \Rightarrow a^{2} + 3 a^{2} - 2 \sqrt{3} a^{2} + a^{2} + x^{2} + 2 a x = 4 a^{2} + 4 a^{2} \Rightarrow x^{2} + 2 a x - 2 \sqrt{3} a^{2} - 3 a^{2} = 0 ...... (1) W h e n {C A}^{2} = {B C}^{2}, {(a - (- \sqrt{3} a))}^{2} + {(a - x)}^{2} = {(a + a)}^{2} + {(a + a)}^{2} \Rightarrow {(a + \sqrt{3} a)}^{2} + {(a - x)}^{2} = {(2 a)}^{2} + {(2 a)}^{2} \Rightarrow a^{2} + 3 a^{2} + 2 \sqrt{3} a^{2} + a^{2} + x^{2} - 2 a x = 4 a^{2} + 4 a^{2} \Rightarrow x^{2} - 2 a x + 2 \sqrt{3} a^{2} - 3 a^{2} = 0 ...... (2) S u b t r a c t i n g e q u a t i o n (1) a n d (2), w e h a v e, 4 a x - 4 \sqrt{3} a^{2} = 0 \Rightarrow 4 a x = 4 \sqrt{3} a^{2} \Rightarrow x = \sqrt{3} a H e n c e, x = \sqrt{3} a$

points (a,a),(-a,-a),and(-root3a,x) r the vertices of the equilateral triangle. find x.

points (a,a),(-a,-a),and(-root3a,x)

r the vertices of the equilateral triangle. find x.