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Properties of Isosceles and Equilateral Triangles

If A 2a,4a an...

Question

If A (2a,4a) and B (2a,6a) are two vertices of a equilateral triangle ABC then the vertex C is given by

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Solution

$A B = \sqrt{(2 a - 2 a)^{2} + (6 a - 4 a)^{2}} = 2 a$
then $B C = 2 a = \sqrt{(x - 2 a)^{2} + (y - 6 a)^{2}}$
also
$A C = 2 a = \sqrt{(x - 2 a)^{2} + (y - 4 a)^{2}}$
on solving,
$x = (\sqrt{3} + 2) a y = 5 a$
$c ((\sqrt{3} + 2) a, 5 a)$

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Q. Assertion :If A(2a, 4a) and B(2a, 6a) are two vertices of a equilateral triangle ABC then the vertex C is given by

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