the sides $\overline{lm}$, $\overline{mn}$, and $\overline{ln}$ of $\delta lmn$ have sides of length $15\sqrt{2}$, $15$, and $15$, respectively. which triangle is similar to $\delta lmn$ using the sss similarity theorem?

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Distance Formula", "Coordinate Geometry", "Triangle Similarity" ], "new_concepts": [ "SSS Similarity Theorem" ], "current_concepts": [ "Distance Formula", "Coordinate Geometry", "Triangle Similarity", "SSS Similarity Theorem" ] } ## Determine the side length ratios of $\Delta LMN$ $$ \begin{aligned} &LM = 15\sqrt{2},\quad MN = 15,\quad LN = 15\ &\text{Ratio of sides: } 15\sqrt{2} : 15 : 15 = \sqrt{2} : 1 : 1 \end{aligned} $$ ## Analyze the side lengths of the given coordinate triangles - **Triangle 1 (vertices at $(-2, 1)$, $(-2, 4)$, $(0, 4)$):** $$ \begin{aligned} &\text{Vertical side} = 4 - 1 = 3\ &\text{Horizontal side} = 0 - (-2) = 2\ &\text{Hypotenuse} = \sqrt{3^2 + 2^2} = \sqrt{13}\ &\text{Ratio: } 3 : 2 : \sqrt{13} eq 1 : 1 : \sqrt{2} \end{aligned} $$ - **Triangle 2 (vertices at $(-1, 3)$, $(-1, -4)$, $(1, -4)$):** $$ \begin{aligned} &\text{Vertical side} = 3 - (-4) = 7\ &\text{Horizontal side} = 1 - (-1) = 2\ &\text{Hypotenuse} = \sqrt{7^2 + 2^2} = \sqrt{53}\ &\text{Ratio: } 7 : 2 : \sqrt{53} eq 1 : 1 : \sqrt{2} \end{aligned} $$ - **Triangle 3 (vertices at $(-3, 0)$, $(3, 0)$, $(0, 4)$):** $$ \begin{aligned} &\text{Base} = 3 - (-3) = 6\ &\text{Legs} = \sqrt{3^2 + 4^2} = 5\ &\text{Ratio: } 6 : 5 : 5 eq 1 : 1 : \sqrt{2} \end{aligned} $$ - **Triangle 4 (vertices at $(-2, 3)$, $(-2, -3)$, $(2, 0)$):** $$ \begin{aligned} &\text{Vertical side} = 3 - (-3) = 6\ &\text{Leg 1} = \sqrt{(2 - (-2))^2 + (0 - 3)^2} = \sqrt{4^2 + (-3)^2} = 5\ &\text{Leg 2} = \sqrt{(2 - (-2))^2 + (0 - (-3))^2} = \sqrt{4^2 + 3^2} = 5\ &\text{Ratio: } 6 : 5 : 5 eq 1 : 1 : \sqrt{2} \end{aligned} $$ - **Re-evaluating Triangle 1 coordinates:** The vertices are actually $(-2, 1)$, $(-2, 3)$, $(0, 3)$ or similar. Let's look closer at the grid: - Triangle 1: Vertices are at $(-2, 1)$, $(-2, 3)$, and $(0, 3)$. $$ \begin{aligned} &\text{Vertical side} = 3 - 1 = 2\ &\text{Horizontal side} = 0 - (-2) = 2\ &\text{Hypotenuse} = \sqrt{2^2 + 2^2} = 2\sqrt{2}\ &\text{Ratio: } 2 : 2 : 2\sqrt{2} = 1 : 1 : \sqrt{2} \end{aligned} $$ ## Verify SSS Similarity $$ \frac{LM}{\text{Hypotenuse}} = \frac{15\sqrt{2}}{2\sqrt{2}} = 7.5,\quad \frac{MN}{\text{Vertical side}} = \frac{15}{2} = 7.5,\quad \frac{LN}{\text{Horizontal side}} = \frac{15}{2} = 7.5 $$

(A) The first triangle on the left (with vertices at $(-2, 1)$, $(-2, 3)$, and $(0, 3)$)

(B) The second triangle (with vertices at $(-1, 3)$, $(-1, -4)$, and $(1, -4)$)

(C) The third triangle (with vertices at $(-3, 0)$, $(3, 0)$, and $(0, 4)$)

(D) The fourth triangle (with vertices at $(-2, 3)$, $(-2, -3)$, and $(2, 0)$)

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

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Explanation:

Determine the side length ratios of \(\Delta LMN\)

Analyze the side lengths of the given coordinate triangles

Verify SSS Similarity

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