Question

triangle \\(abc\\) is similar to triangle \\(lmn\\), which is formed using \\(\overline{lm}\\), as shown.

what could be the coordinates of point \\(n\\)?

(3,4)
(4,-3)
(-3,4)
(4,3)

Explanation:

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<reasoning>

Determine the coordinates and side lengths of triangle ABC

\[

$$\begin{aligned} &A = (5, 2),\quad B = (5, -2),\quad C = (2, 1)\\ &AB = \sqrt{(5-5)^2 + (2 - (-2))^2} = 4\\ &BC = \sqrt{(5-2)^2 + (-2-1)^2} = \sqrt{9 + 9} = 3\sqrt{2}\\ &AC = \sqrt{(5-2)^2 + (2-1)^2} = \sqrt{9 + 1} = \sqrt{10} \end{aligned}$$

\]

Determine the scale factor using segment LM

\[

$$\begin{aligned} &L = (-5, 6),\quad M = (-5, -6)\\ &LM = \sqrt{(-5 - (-5))^2 + (6 - (-6))^2} = 12\\ &k = \frac{LM}{AB} = \frac{12}{4} = 3 \end{aligned}$$

\]

Find the coordinates of point N

\[

$$\begin{aligned} &LN = 3 \times AC = 3\sqrt{10}\\ &MN = 3 \times BC = 9\sqrt{2}\\ &(x_N - (-5))^2 + (y_N - 6)^2 = 90 \implies (x_N + 5)^2 + (y_N - 6)^2 = 90\\ &(x_N - (-5))^2 + (y_N - (-6))^2 = 162 \implies (x_N + 5)^2 + (y_N + 6)^2 = 162\\ &\text{Subtracting the equations: } 24y_N = 72 \implies y_N = 3\\ &(x_N + 5)^2 + (3 - 6)^2 = 90 \implies (x_N + 5)^2 = 81 \implies x_N + 5 = \pm 9\\ &x_N = 4 \quad (\text{since } x_N > -5 \text{ from the options})\\ &N = (4, 3) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(3,4)</mcq-option>
<mcq-option>(4,-3)</mcq-option>
<mcq-option>(-3,4)</mcq-option>
<mcq-correct>(4,3)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Determine the coordinates and side lengths of triangle ABC

\[

$$\begin{aligned} &A = (5, 2),\quad B = (5, -2),\quad C = (2, 1)\\ &AB = \sqrt{(5-5)^2 + (2 - (-2))^2} = 4\\ &BC = \sqrt{(5-2)^2 + (-2-1)^2} = \sqrt{9 + 9} = 3\sqrt{2}\\ &AC = \sqrt{(5-2)^2 + (2-1)^2} = \sqrt{9 + 1} = \sqrt{10} \end{aligned}$$

\]

Determine the scale factor using segment LM

\[

$$\begin{aligned} &L = (-5, 6),\quad M = (-5, -6)\\ &LM = \sqrt{(-5 - (-5))^2 + (6 - (-6))^2} = 12\\ &k = \frac{LM}{AB} = \frac{12}{4} = 3 \end{aligned}$$

\]

Find the coordinates of point N

\[

$$\begin{aligned} &LN = 3 \times AC = 3\sqrt{10}\\ &MN = 3 \times BC = 9\sqrt{2}\\ &(x_N - (-5))^2 + (y_N - 6)^2 = 90 \implies (x_N + 5)^2 + (y_N - 6)^2 = 90\\ &(x_N - (-5))^2 + (y_N - (-6))^2 = 162 \implies (x_N + 5)^2 + (y_N + 6)^2 = 162\\ &\text{Subtracting the equations: } 24y_N = 72 \implies y_N = 3\\ &(x_N + 5)^2 + (3 - 6)^2 = 90 \implies (x_N + 5)^2 = 81 \implies x_N + 5 = \pm 9\\ &x_N = 4 \quad (\text{since } x_N > -5 \text{ from the options})\\ &N = (4, 3) \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(3,4)</mcq-option>
<mcq-option>(4,-3)</mcq-option>
<mcq-option>(-3,4)</mcq-option>
<mcq-correct>(4,3)</mcq-correct>
</answer>

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