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)

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)

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Coordinate Similarity", "Geometric Transformations" ], "current_concepts": [ "Coordinate Geometry", "Triangle Similarity", "Scale Factor" ] } ## 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 $$ \begin{aligned} &LN = 3 \times AC = 3\sqrt{10}\ &MN = 3 \times BC = 9\sqrt{2}\ &(x_N - (-5))^2 + (y_N - 6)^2 = 90 \implies (x_N + 5)^2 + (y_N - 6)^2 = 90\ &(x_N - (-5))^2 + (y_N - (-6))^2 = 162 \implies (x_N + 5)^2 + (y_N + 6)^2 = 162\ &\text{Subtracting the equations: } 24y_N = 72 \implies y_N = 3\ &(x_N + 5)^2 + (3 - 6)^2 = 90 \implies (x_N + 5)^2 = 81 \implies x_N + 5 = \pm 9\ &x_N = 4 \quad (\text{since } x_N > -5 \text{ from the options})\ &N = (4, 3) \end{aligned} $$

(3,4)

(4,-3)

(-3,4)

(4,3)

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Triangle Similarity" ] }

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QUESTION IMAGE

Question

Explanation:

Determine the coordinates and side lengths of triangle ABC

Determine the scale factor using segment LM

Find the coordinates of point N

Answer: