segment $hi$ in the coordinate plane has endpoints $h(5,4)$ and $i(3,4)$. the line is dilated by a scale factor of $\frac{1}{2}$ with point $i$ as the center of dilation, resulting in the image $\overline{hi}$. which image shows $\overline{hi}$?

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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": [ "Geometric Transformations" ], "new_concepts": [ "Dilation in Coordinate Plane" ], "current_concepts": [ "Geometric Transformations", "Dilation in Coordinate Plane" ] } ## Calculate the coordinates of the dilated endpoints $$ \begin{aligned} &H = (5, 4), \quad I = (3, 4), \quad \text{Center of dilation } C = I = (3, 4), \quad \text{Scale factor } k = \frac{1}{2}\ &I' = C = (3, 4)\ &H' = C + k(H - C) = (3, 4) + \frac{1}{2}((5, 4) - (3, 4)) = (3, 4) + \frac{1}{2}(2, 0) = (4, 4) \end{aligned} $$ ## Identify the correct graph $$ \begin{aligned} &I' = (3, 4)\ &H' = (4, 4) \end{aligned} $$ The third graph shows a horizontal segment with endpoints at $(3, 4)$ and $(4, 4)$.

(A) Segment with endpoints at $I'(3, 4)$ and $H'(5, 5)$

(B) Segment with endpoints at $I'(3, 4)$ and $H'(7, 4)$

(C) Segment with endpoints at $I'(3, 4)$ and $H'(4, 4)$

(D) Segment with endpoints at $I'(3, 4)$ and $H'(4, 6)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Geometric Transformations" ] }

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Explanation:

Calculate the coordinates of the dilated endpoints

Identify the correct graph

Answer: