Final answer:
To find the fourth vertex of a rhombus given three vertices, we make use of the property of rhombus where diagonals are perpendicular bisectors of each other. The midpoint of the known diagonal is the same as the midpoint of the unknown diagonal. Solving these will provide the coordinates of the missing point.
Step-by-step explanation:
Given the points A(-5,5), B(0,0), and C(7,1), we need to find the fourth point (D) in order to form a parallelogram. What makes this a special case is that the parallelogram is noted to be a rhombus. A rhombus is a type of parallelogram where all four sides are of equal length, thus its diagonals are perpendicular bisectors of each other which also means they bisect each other into two equal parts.
Given the diagonal points A(-5,5) and C(7,1), we find the midpoint, which will be the same as the midpoint of BD. The midpoint formula is [(x₁+x₂)/2 , (y₁+y₂)/2]. Thus, the midpoint M(x,y) of AC is M[(7-5)/2 , (1+5)/2] => M[1, 3].
To find the coordinates of point D, we apply the same logic to find midpoint of AB and MC. Midpoint of AB: M₁[(-5+0)/2 , (5+0)/2] => M₁[-2.5, 2.5]. Equating this to MC gives us a system of equations which can be solved to yield the coordinates of point D, which should form a rhombus ABCD when connected with the given points.
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