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Show that the points A(1,2),B(1,6), C(1 ...

Show that the points A(1,2),B(1,6), `C(1 + 2sqrt(3),4) ` are the
vertices of an equilateral triangle .

Text Solution

Verified by Experts

`A(1,2),B(1,6),C(1+2sqrt(3),4)`
By distance formula.
`AB=sqrt((1-1)^(2)+(6-2)^(2))`
`:.AB=sqrt(0^(2)+4^(2))`
`:.AB=sqrt(4^(2))`
`:.AB=4`
`BC=sqrt((1+2sqrt(3-1)^(2)+(4-6)^(2))`
`:.BC=sqrt((2sqrt(3)^(2)+(-2)^(2))`
`:.BC=sqrt(12+4)`
`:.BC=sqrt(16)`
`:.BC=4`
`AC=sqrt((1+2sqrt(3)-1)^(2)+(4-2)^(2))`
`:.AC=sqrt((2sqrt(3)^(2)+(2)^(2))`
`:.AC=sqrt(12+4)`
`:.AC=sqrt(16)`
`:.AC=4`
`:.AB=BC=AC`
`:.DeltaABC` is an equilateral triangle.
`A(1,2),B(1,6),C(1+2sqrt(3),4)` are vertices of an equilateral triangle.
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