the sides $\overline{lm}$, $\overline{mn}$, and $\overline{ln}$ of $\triangle lmn$ have lengths of 3, 4, and 5, respectively. which triangle is congruent to $\triangle lmn$?

Question

Accepted Answer

{ "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 Congruence" ], "new_concepts": [], "current_concepts": [ "Distance Formula", "Coordinate Geometry", "Triangle Congruence" ] } ## Determine the side lengths of the target triangle $$ \begin{aligned} &LM = 3\ &MN = 4\ &LN = 5 \end{aligned} $$ ## Analyze the side lengths of each option - **Option 1:** Vertices at $(-2, 0)$, $(-2, 4)$, and $(2, 4)$. $$ \begin{aligned} &\text{Vertical side} = |4 - 0| = 4\ &\text{Horizontal side} = |2 - (-2)| = 4\ &\text{Hypotenuse} = \sqrt{4^2 + 4^2} = 4\sqrt{2} \approx 5.66 \end{aligned} $$ - **Option 2:** Vertices at $(-1, 3)$, $(-1, -2)$, and $(3, -2)$. $$ \begin{aligned} &\text{Vertical side} = |3 - (-2)| = 5\ &\text{Horizontal side} = |3 - (-1)| = 4\ &\text{Hypotenuse} = \sqrt{5^2 + 4^2} = \sqrt{41} \approx 6.40 \end{aligned} $$ - **Option 3:** Vertices at $(-1, 3)$, $(-1, -1)$, and $(2, -1)$. $$ \begin{aligned} &\text{Vertical side} = |3 - (-1)| = 4\ &\text{Horizontal side} = |2 - (-1)| = 3\ &\text{Hypotenuse} = \sqrt{4^2 + 3^2} = 5 \end{aligned} $$ - **Option 4:** Vertices at $(0, 3)$, $(0, -4)$, and $(3, -4)$. $$ \begin{aligned} &\text{Vertical side} = |3 - (-4)| = 7\ &\text{Horizontal side} = |3 - 0| = 3\ &\text{Hypotenuse} = \sqrt{7^2 + 3^2} = \sqrt{58} \approx 7.62 \end{aligned} $$ ## Identify the congruent triangle $$ \text{Option 3 has side lengths } 3, 4, \text{ and } 5 \implies \text{Congruent by SSS} $$

(A) First option (triangle with vertices at $(-2, 0)$, $(-2, 4)$, $(2, 4)$)

(B) Second option (triangle with vertices at $(-1, 3)$, $(-1, -2)$, $(3, -2)$)

(C) Third option (triangle with vertices at $(-1, 3)$, $(-1, -1)$, $(2, -1)$)

(D) Fourth option (triangle with vertices at $(0, 3)$, $(0, -4)$, $(3, -4)$)

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

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Determine the side lengths of the target triangle

Analyze the side lengths of each option

Identify the congruent triangle

Answer: