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प्रश्न
Show that the points A(1, 2), B(1, 6), C(1 + 2`sqrt3`, 4) are vertices of an equilateral triangle.
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उत्तर
The given points are A(1, 2), B(1, 6), C(1 + 2`sqrt3`, 4).
`"Distance between" = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
By distance formula,
AB = `sqrt((1 - 1)^2 + (6 - 2)^2)`
∴ AB = `sqrt((0)^2 + (4)^2)`
∴ AB = `sqrt(0 + 16)`
∴ AB = `sqrt(16)`
∴ AB = 4 ...(1)
BC = `sqrt((1 + 2sqrt3 - 1)^2 + (4 - 6)^2)`
∴ BC = `sqrt((2sqrt3)^2 + (-2)^2)`
∴ BC = `sqrt(12 + 4)`
∴ BC = `sqrt(16)`
∴ BC = 4 ...(2)
AC = `sqrt((1 + 2sqrt3 - 1)^2 + (4 - 2)^2)`
∴ AC = `sqrt((2sqrt3)^2 + (2)^2)`
∴ AC = `sqrt(12 + 4)`
∴ AC = `sqrt(16)`
∴ AC = 4 ...(3)
From (1), (2) and (3)
∴ AB = BC = AC = 4
Since, all the sides of equilateral triangle are congruent.
∴ ΔABC is an equilateral triangle.
The points A, B and C are the vertices of an equilateral triangle.
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Alia and Shagun are friends living on the same street in Patel Nagar. Shagun's house is at the intersection of one street with another street on which there is a library. They both study in the same school and that is not far from Shagun's house. Suppose the school is situated at the point O, i.e., the origin, Alia's house is at A. Shagun's house is at B and library is at C. |
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