Join the IDNLearn.com community and get your questions answered by experts. Join our platform to receive prompt and accurate responses from experienced professionals in various fields.

The side lengths of triangle [tex]$ABC$[/tex] are 3, 4, and 5. Which set of ordered pairs forms a triangle that is congruent to triangle [tex]$ABC$[/tex]?

A. [tex]$(-3,1),(-3,5),(0,5)$[/tex]
B. [tex]$(1,-2),(1,3),(4,3)$[/tex]
C. [tex]$(6,-3),(1,-3),(6,1)$[/tex]
D. [tex]$(1,3),(1,4),(5,1)$[/tex]


Sagot :

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.