Answer :
To determine which set of ordered pairs form a triangle that is congruent to triangle [tex]\(ABC\)[/tex] with side lengths 3, 4, and 5, we should calculate the distances between each pair of points in the sets provided. Let's go through each set step-by-step and compare their side lengths to those of triangle [tex]\(ABC\)[/tex].
### Set 1: [tex]\((-3,1)\)[/tex], [tex]\((-3,5)\)[/tex], [tex]\((0,5)\)[/tex]
1. Calculate the distance between [tex]\((-3,1)\)[/tex] and [tex]\((-3,5)\)[/tex]:
[tex]\[ d = \sqrt{((-3) - (-3))^2 + (5 - 1)^2} = \sqrt{0^2 + 4^2} = 4 \][/tex]
2. Calculate the distance between [tex]\((-3,5)\)[/tex] and [tex]\((0,5)\)[/tex]:
[tex]\[ d = \sqrt{(0 - (-3))^2 + (5 - 5)^2} = \sqrt{3^2 + 0^2} = 3 \][/tex]
3. Calculate the distance between [tex]\((0,5)\)[/tex] and [tex]\((-3,1)\)[/tex]:
[tex]\[ d = \sqrt{(0 - (-3))^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = 5 \][/tex]
The side lengths of this set are [tex]\(3\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex], which match triangle [tex]\(ABC\)[/tex].
### Set 2: [tex]\((1,-2)\)[/tex], [tex]\((1,3)\)[/tex], [tex]\((4,3)\)[/tex]
1. Calculate the distance between [tex]\((1,-2)\)[/tex] and [tex]\((1,3)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 1)^2 + (3 - (-2))^2} = \sqrt{0^2 + 5^2} = 5 \][/tex]
2. Calculate the distance between [tex]\((1,3)\)[/tex] and [tex]\((4,3)\)[/tex]:
[tex]\[ d = \sqrt{(4 - 1)^2 + (3 - 3)^2} = \sqrt{3^2 + 0^2} = 3 \][/tex]
3. Calculate the distance between [tex]\((4,3)\)[/tex] and [tex]\((1,-2)\)[/tex]:
[tex]\[ d = \sqrt{(4 - 1)^2 + (3 - (-2))^2} = \sqrt{3^2 + 5^2} = \sqrt{34} \][/tex]
The side lengths of this set are [tex]\(3\)[/tex], [tex]\(5\)[/tex], and [tex]\(\sqrt{34}\)[/tex], which do not match triangle [tex]\(ABC\)[/tex].
### Set 3: [tex]\((6,-3)\)[/tex], [tex]\((1,-3)\)[/tex], [tex]\((6,1)\)[/tex]
1. Calculate the distance between [tex]\((6,-3)\)[/tex] and [tex]\((1,-3)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 6)^2 + (-3 - (-3))^2} = \sqrt{(-5)^2 + 0^2} = 5 \][/tex]
2. Calculate the distance between [tex]\((1,-3)\)[/tex] and [tex]\((6,1)\)[/tex]:
[tex]\[ d = \sqrt{(6 - 1)^2 + (1 - (-3))^2} = \sqrt{5^2 + 4^2} = \sqrt{41} \][/tex]
3. Calculate the distance between [tex]\((6,1)\)[/tex] and [tex]\((6,-3)\)[/tex]:
[tex]\[ d = \sqrt{(6 - 6)^2 + (1 - (-3))^2} = \sqrt{0^2 + 4^2} = 4 \][/tex]
The side lengths of this set are [tex]\(4\)[/tex], [tex]\(5\)[/tex], and [tex]\(\sqrt{41}\)[/tex], which do not match triangle [tex]\(ABC\)[/tex].
### Set 4: [tex]\((1,3)\)[/tex], [tex]\((1,4)\)[/tex], [tex]\((5,1)\)[/tex]
1. Calculate the distance between [tex]\((1,3)\)[/tex] and [tex]\((1,4)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 1)^2 + (4 - 3)^2} = \sqrt{0^2 + 1^2} = 1 \][/tex]
2. Calculate the distance between [tex]\((1,4)\)[/tex] and [tex]\((5,1)\)[/tex]:
[tex]\[ d = \sqrt{(5 - 1)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = 5 \][/tex]
3. Calculate the distance between [tex]\((5,1)\)[/tex] and [tex]\((1,3)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 5)^2 + (3 - 1)^2} = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} \][/tex]
The side lengths of this set are [tex]\(1\)[/tex], [tex]\(5\)[/tex], and [tex]\(\sqrt{20}\)[/tex], which do not match triangle [tex]\(ABC\)[/tex].
### Conclusion
The set of ordered pairs that form a triangle congruent to triangle [tex]\(ABC\)[/tex] is:
[tex]\[ (-3,1), (-3,5), (0,5) \][/tex]
This is Set 1.
### Set 1: [tex]\((-3,1)\)[/tex], [tex]\((-3,5)\)[/tex], [tex]\((0,5)\)[/tex]
1. Calculate the distance between [tex]\((-3,1)\)[/tex] and [tex]\((-3,5)\)[/tex]:
[tex]\[ d = \sqrt{((-3) - (-3))^2 + (5 - 1)^2} = \sqrt{0^2 + 4^2} = 4 \][/tex]
2. Calculate the distance between [tex]\((-3,5)\)[/tex] and [tex]\((0,5)\)[/tex]:
[tex]\[ d = \sqrt{(0 - (-3))^2 + (5 - 5)^2} = \sqrt{3^2 + 0^2} = 3 \][/tex]
3. Calculate the distance between [tex]\((0,5)\)[/tex] and [tex]\((-3,1)\)[/tex]:
[tex]\[ d = \sqrt{(0 - (-3))^2 + (5 - 1)^2} = \sqrt{3^2 + 4^2} = 5 \][/tex]
The side lengths of this set are [tex]\(3\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex], which match triangle [tex]\(ABC\)[/tex].
### Set 2: [tex]\((1,-2)\)[/tex], [tex]\((1,3)\)[/tex], [tex]\((4,3)\)[/tex]
1. Calculate the distance between [tex]\((1,-2)\)[/tex] and [tex]\((1,3)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 1)^2 + (3 - (-2))^2} = \sqrt{0^2 + 5^2} = 5 \][/tex]
2. Calculate the distance between [tex]\((1,3)\)[/tex] and [tex]\((4,3)\)[/tex]:
[tex]\[ d = \sqrt{(4 - 1)^2 + (3 - 3)^2} = \sqrt{3^2 + 0^2} = 3 \][/tex]
3. Calculate the distance between [tex]\((4,3)\)[/tex] and [tex]\((1,-2)\)[/tex]:
[tex]\[ d = \sqrt{(4 - 1)^2 + (3 - (-2))^2} = \sqrt{3^2 + 5^2} = \sqrt{34} \][/tex]
The side lengths of this set are [tex]\(3\)[/tex], [tex]\(5\)[/tex], and [tex]\(\sqrt{34}\)[/tex], which do not match triangle [tex]\(ABC\)[/tex].
### Set 3: [tex]\((6,-3)\)[/tex], [tex]\((1,-3)\)[/tex], [tex]\((6,1)\)[/tex]
1. Calculate the distance between [tex]\((6,-3)\)[/tex] and [tex]\((1,-3)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 6)^2 + (-3 - (-3))^2} = \sqrt{(-5)^2 + 0^2} = 5 \][/tex]
2. Calculate the distance between [tex]\((1,-3)\)[/tex] and [tex]\((6,1)\)[/tex]:
[tex]\[ d = \sqrt{(6 - 1)^2 + (1 - (-3))^2} = \sqrt{5^2 + 4^2} = \sqrt{41} \][/tex]
3. Calculate the distance between [tex]\((6,1)\)[/tex] and [tex]\((6,-3)\)[/tex]:
[tex]\[ d = \sqrt{(6 - 6)^2 + (1 - (-3))^2} = \sqrt{0^2 + 4^2} = 4 \][/tex]
The side lengths of this set are [tex]\(4\)[/tex], [tex]\(5\)[/tex], and [tex]\(\sqrt{41}\)[/tex], which do not match triangle [tex]\(ABC\)[/tex].
### Set 4: [tex]\((1,3)\)[/tex], [tex]\((1,4)\)[/tex], [tex]\((5,1)\)[/tex]
1. Calculate the distance between [tex]\((1,3)\)[/tex] and [tex]\((1,4)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 1)^2 + (4 - 3)^2} = \sqrt{0^2 + 1^2} = 1 \][/tex]
2. Calculate the distance between [tex]\((1,4)\)[/tex] and [tex]\((5,1)\)[/tex]:
[tex]\[ d = \sqrt{(5 - 1)^2 + (1 - 4)^2} = \sqrt{4^2 + (-3)^2} = 5 \][/tex]
3. Calculate the distance between [tex]\((5,1)\)[/tex] and [tex]\((1,3)\)[/tex]:
[tex]\[ d = \sqrt{(1 - 5)^2 + (3 - 1)^2} = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} \][/tex]
The side lengths of this set are [tex]\(1\)[/tex], [tex]\(5\)[/tex], and [tex]\(\sqrt{20}\)[/tex], which do not match triangle [tex]\(ABC\)[/tex].
### Conclusion
The set of ordered pairs that form a triangle congruent to triangle [tex]\(ABC\)[/tex] is:
[tex]\[ (-3,1), (-3,5), (0,5) \][/tex]
This is Set 1.
