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To determine whether the given points form the vertices of a rhombus, we need to show that all sides of each quadrilateral are of equal length. Let's begin by analyzing each set of points step-by-step.
### Part a: Quadrilateral with vertices (2,3), (5,8), (0,5), (-3,0)
1. Calculate the distances between each pair of consecutive vertices:
- Distance between (2, 3) and (5, 8):
[tex]\[ d_1 = \sqrt{(5 - 2)^2 + (8 - 3)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
- Distance between (5, 8) and (0, 5):
[tex]\[ d_2 = \sqrt{(0 - 5)^2 + (5 - 8)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \][/tex]
- Distance between (0, 5) and (-3, 0):
[tex]\[ d_3 = \sqrt{(0 - (-3))^2 + (5 - 0)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
- Distance between (-3, 0) and (2, 3):
[tex]\[ d_4 = \sqrt{(2 - (-3))^2 + (3 - 0)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \][/tex]
2. Evaluate whether all four distances are equal:
[tex]\[ d_1 = d_2 = d_3 = d_4 = \sqrt{34} \][/tex]
Since the distances between each pair of consecutive vertices are all equal, the quadrilateral formed by the points [tex]\((2, 3)\)[/tex], [tex]\((5, 8)\)[/tex], [tex]\((0, 5)\)[/tex], and [tex]\((-3, 0)\)[/tex] is a rhombus.
### Part b: Quadrilateral with vertices (-3,-2), (-2,3), (3,4), (2,1)
1. Calculate the distances between each pair of consecutive vertices:
- Distance between (-3, -2) and (-2, 3):
[tex]\[ d_1 = \sqrt{(-2 - (-3))^2 + (3 - (-2))^2} = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26} \][/tex]
- Distance between (-2, 3) and (3, 4):
[tex]\[ d_2 = \sqrt{(3 - (-2))^2 + (4 - 3)^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \][/tex]
- Distance between (3, 4) and (2, 1):
[tex]\[ d_3 = \sqrt{(2 - 3)^2 + (1 - 4)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \][/tex]
- Distance between (2, 1) and (-3, -2):
[tex]\[ d_4 = \sqrt{(2 - (-3))^2 + (1 - (-2))^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \][/tex]
2. Evaluate whether all four distances are equal:
[tex]\[ d_1 = \sqrt{26}, \quad d_2 = \sqrt{26}, \quad d_3 = \sqrt{10}, \quad d_4 = \sqrt{34} \][/tex]
Since the distances between the vertices are not all equal, the quadrilateral formed by the points [tex]\((-3, -2)\)[/tex], [tex]\((-2, 3)\)[/tex], [tex]\((3, 4)\)[/tex], and [tex]\((2, 1)\)[/tex] does not have all sides of equal length and therefore is not a rhombus.
### Conclusion:
- The quadrilateral with vertices [tex]\((2, 3)\)[/tex], [tex]\((5, 8)\)[/tex], [tex]\((0, 5)\)[/tex], and [tex]\((-3, 0)\)[/tex] is a rhombus.
- The quadrilateral with vertices [tex]\((-3, -2)\)[/tex], [tex]\((-2, 3)\)[/tex], [tex]\((3, 4)\)[/tex], and [tex]\((2, 1)\)[/tex] is not a rhombus.
### Part a: Quadrilateral with vertices (2,3), (5,8), (0,5), (-3,0)
1. Calculate the distances between each pair of consecutive vertices:
- Distance between (2, 3) and (5, 8):
[tex]\[ d_1 = \sqrt{(5 - 2)^2 + (8 - 3)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
- Distance between (5, 8) and (0, 5):
[tex]\[ d_2 = \sqrt{(0 - 5)^2 + (5 - 8)^2} = \sqrt{(-5)^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34} \][/tex]
- Distance between (0, 5) and (-3, 0):
[tex]\[ d_3 = \sqrt{(0 - (-3))^2 + (5 - 0)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \][/tex]
- Distance between (-3, 0) and (2, 3):
[tex]\[ d_4 = \sqrt{(2 - (-3))^2 + (3 - 0)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \][/tex]
2. Evaluate whether all four distances are equal:
[tex]\[ d_1 = d_2 = d_3 = d_4 = \sqrt{34} \][/tex]
Since the distances between each pair of consecutive vertices are all equal, the quadrilateral formed by the points [tex]\((2, 3)\)[/tex], [tex]\((5, 8)\)[/tex], [tex]\((0, 5)\)[/tex], and [tex]\((-3, 0)\)[/tex] is a rhombus.
### Part b: Quadrilateral with vertices (-3,-2), (-2,3), (3,4), (2,1)
1. Calculate the distances between each pair of consecutive vertices:
- Distance between (-3, -2) and (-2, 3):
[tex]\[ d_1 = \sqrt{(-2 - (-3))^2 + (3 - (-2))^2} = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26} \][/tex]
- Distance between (-2, 3) and (3, 4):
[tex]\[ d_2 = \sqrt{(3 - (-2))^2 + (4 - 3)^2} = \sqrt{5^2 + 1^2} = \sqrt{25 + 1} = \sqrt{26} \][/tex]
- Distance between (3, 4) and (2, 1):
[tex]\[ d_3 = \sqrt{(2 - 3)^2 + (1 - 4)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \][/tex]
- Distance between (2, 1) and (-3, -2):
[tex]\[ d_4 = \sqrt{(2 - (-3))^2 + (1 - (-2))^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \][/tex]
2. Evaluate whether all four distances are equal:
[tex]\[ d_1 = \sqrt{26}, \quad d_2 = \sqrt{26}, \quad d_3 = \sqrt{10}, \quad d_4 = \sqrt{34} \][/tex]
Since the distances between the vertices are not all equal, the quadrilateral formed by the points [tex]\((-3, -2)\)[/tex], [tex]\((-2, 3)\)[/tex], [tex]\((3, 4)\)[/tex], and [tex]\((2, 1)\)[/tex] does not have all sides of equal length and therefore is not a rhombus.
### Conclusion:
- The quadrilateral with vertices [tex]\((2, 3)\)[/tex], [tex]\((5, 8)\)[/tex], [tex]\((0, 5)\)[/tex], and [tex]\((-3, 0)\)[/tex] is a rhombus.
- The quadrilateral with vertices [tex]\((-3, -2)\)[/tex], [tex]\((-2, 3)\)[/tex], [tex]\((3, 4)\)[/tex], and [tex]\((2, 1)\)[/tex] is not a rhombus.
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