Question

the side lengths of triangle \\(rst\\) are 5, 12, and 13. which set of ordered pairs form a triangle that is congruent to triangle \\(rst\\)?

\\((-3,-2), (-3,11), (9,-2)\\)
\\((1,-3), (6,-3), (6,10)\\)
\\((2,-1), (7,-1), (2,11)\\)
\\((5,1), (1,13), (12,1)\\)

Explanation:

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<reasoning>

Calculate side lengths for the first option

\[

$$\begin{aligned} &A(-3,-2), \quad B(-3,11), \quad C(9,-2) \\ &d(A,B) = \sqrt{(-3 - (-3))^2 + (11 - (-2))^2} = \sqrt{0 + 13^2} = 13 \\ &d(A,C) = \sqrt{(9 - (-3))^2 + (-2 - (-2))^2} = \sqrt{12^2 + 0} = 12 \\ &d(B,C) = \sqrt{(9 - (-3))^2 + (-2 - 11)^2} = \sqrt{12^2 + (-13)^2} = \sqrt{144 + 169} = \sqrt{313} eq 5 \end{aligned}$$

\]

Calculate side lengths for the second option

\[

$$\begin{aligned} &A(1,-3), \quad B(6,-3), \quad C(6,10) \\ &d(A,B) = \sqrt{(6-1)^2 + (-3 - (-3))^2} = \sqrt{5^2 + 0} = 5 \\ &d(B,C) = \sqrt{(6-6)^2 + (10 - (-3))^2} = \sqrt{0 + 13^2} = 13 \\ &d(A,C) = \sqrt{(6-1)^2 + (10 - (-3))^2} = \sqrt{5^2 + 13^2} = \sqrt{25 + 169} = \sqrt{194} eq 12 \end{aligned}$$

\]

Calculate side lengths for the third option

\[

$$\begin{aligned} &A(2,-1), \quad B(7,-1), \quad C(2,11) \\ &d(A,B) = \sqrt{(7-2)^2 + (-1 - (-1))^2} = \sqrt{5^2 + 0} = 5 \\ &d(A,C) = \sqrt{(2-2)^2 + (11 - (-1))^2} = \sqrt{0 + 12^2} = 12 \\ &d(B,C) = \sqrt{(2-7)^2 + (11 - (-1))^2} = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(-3,-2),(-3,11),(9,-2)</mcq-option>
<mcq-option>(1,-3),(6,-3),(6,10)</mcq-option>
<mcq-correct>(2,-1),(7,-1),(2,11)</mcq-correct>
<mcq-option>(5,1),(1,13),(12,1)</mcq-option>
</answer>

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Answer:

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<reasoning>

Calculate side lengths for the first option

\[

$$\begin{aligned} &A(-3,-2), \quad B(-3,11), \quad C(9,-2) \\ &d(A,B) = \sqrt{(-3 - (-3))^2 + (11 - (-2))^2} = \sqrt{0 + 13^2} = 13 \\ &d(A,C) = \sqrt{(9 - (-3))^2 + (-2 - (-2))^2} = \sqrt{12^2 + 0} = 12 \\ &d(B,C) = \sqrt{(9 - (-3))^2 + (-2 - 11)^2} = \sqrt{12^2 + (-13)^2} = \sqrt{144 + 169} = \sqrt{313} eq 5 \end{aligned}$$

\]

Calculate side lengths for the second option

\[

$$\begin{aligned} &A(1,-3), \quad B(6,-3), \quad C(6,10) \\ &d(A,B) = \sqrt{(6-1)^2 + (-3 - (-3))^2} = \sqrt{5^2 + 0} = 5 \\ &d(B,C) = \sqrt{(6-6)^2 + (10 - (-3))^2} = \sqrt{0 + 13^2} = 13 \\ &d(A,C) = \sqrt{(6-1)^2 + (10 - (-3))^2} = \sqrt{5^2 + 13^2} = \sqrt{25 + 169} = \sqrt{194} eq 12 \end{aligned}$$

\]

Calculate side lengths for the third option

\[

$$\begin{aligned} &A(2,-1), \quad B(7,-1), \quad C(2,11) \\ &d(A,B) = \sqrt{(7-2)^2 + (-1 - (-1))^2} = \sqrt{5^2 + 0} = 5 \\ &d(A,C) = \sqrt{(2-2)^2 + (11 - (-1))^2} = \sqrt{0 + 12^2} = 12 \\ &d(B,C) = \sqrt{(2-7)^2 + (11 - (-1))^2} = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(-3,-2),(-3,11),(9,-2)</mcq-option>
<mcq-option>(1,-3),(6,-3),(6,10)</mcq-option>
<mcq-correct>(2,-1),(7,-1),(2,11)</mcq-correct>
<mcq-option>(5,1),(1,13),(12,1)</mcq-option>
</answer>

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