Questions: Miguel plotted the three vertices of a triangle. Two of the vertices are shown on the grid below. Which of these could be all three vertices of Miguel's triangle? Select all that apply. (a) (1,1),(1,3),(4,2) (b) (1,1),(3,1),(2,4) (c) (1,3),(3,5),(4,2) (d) (3,1),(2,1),(4,2) (e) (3,1),(2,4),(3,5)

Which of these could be all three vertices of Miguel's triangle? Select all that apply.

Identify the given vertices

The two given vertices are \((1, 3)\) and \((4, 2)\).

Check option (a)

Option (a) is \((1, 1), (1, 3), (4, 2)\). The given vertices \((1, 3)\) and \((4, 2)\) are present. Therefore, this option is a valid triangle.

Check option (b)

Option (b) is \((1, 1), (3, 1), (2, 4)\). Neither of the given vertices \((1, 3)\) and \((4, 2)\) are present. Thus, this is not a valid triangle.

Check option (c)

Option (c) is \((1, 3), (3, 5), (4, 2)\). The given vertices \((1, 3)\) and \((4, 2)\) are present. Thus, this is a valid triangle.

Check option (d)

Option (d) is \((3, 1), (2, 1), (4, 2)\). Only \((4, 2)\) is present. Thus, this is not a valid triangle.

Check option (e)

Option (e) is \((3, 1), (2, 4), (3, 5)\). Neither of the given vertices \((1, 3)\) and \((4, 2)\) are present. Thus, this is not a valid triangle.

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\( \boxed{\text{(a) and (c)}} \)