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Show that points (2-sqrt3 , sqrt3 +1) , ...

Show that points `(2-sqrt3 , sqrt3 +1) , (1,0)` and `(3,2)` form an equilateral triangle .

Text Solution

Verified by Experts

Let A`(2-sqrt3 , sqrt3 + 1)` , B (1, 0) and C(3,2) be the given points .
`AB = sqrt((1-2 + sqrt3)^(2) + (0-(sqrt3 +1))^(2))`
`= sqrt((sqrt3-1)^(2) + (-1 - sqrt3)^(2))`
AB = `sqrt8` units .
`BC = sqrt((3-1)^(2) + (2-sqrt3)^(2)) = sqrt8` units .
AC = `sqrt((3-(2-sqrt3))^(2) + (2-(sqrt3 +1)^(2)))`
`= sqrt((1+sqrt3)^(2) + (1-sqrt3)^(2)) = sqrt8` units.
`therefore` AB = BC = AC = `sqrt8` units .
Hence , the given points form an equilateral triangle .
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