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in 3D Coordinate Geometry by (55.0k points)
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Show that the points A(1, -1, -5), b(3, 1,3) and C(9, 1, -3) are the vertices of an equilateral triangle.

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To prove: Points A, B, C form equilateral triangle. 

Formula: The distance between two points (x1,y1,z1) and (x2,y2,z2) is given by

D = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Here, 

(x1,y1,z1)= (1, -1, -5) 

(x2,y2,z2)= (3, 1,3) 

(x3,y3,z3)= (9, 1, -3)

Length AB = \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)

Length BC = \(\sqrt{(x_3-x_2)^2+(y_3-y_2)^2+(z_3-z_2)^2}\)

Length AC = \(\sqrt{(x_3-x_1)^2+(y_3-y_1)^2+(z_3-z_1)^2}\)

Hence, AB = BC = AC

Therefore, points A,B,C make an equilateral triangle.

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