kite efgh is inscribed in a rectangle such that f and h are midpoints and eg is parallel to the side of the rectangle.

which statement describes how the location of segment eg affects the area of efgh?

- the area of efgh is $\frac{1}{4}$ of the area of the rectangle if e and g are not midpoints.
- the area of efgh is $\frac{1}{2}$ of the area of the rectangle only if e and g are midpoints.
- the area of efgh is always $\frac{1}{2}$ of the area of the rectangle.
- the area of efgh is always $\frac{1}{4}$ of the area of the rectangle.

Question

kite efgh is inscribed in a rectangle such that f and h are midpoints and eg is parallel to the side of the rectangle.

which statement describes how the location of segment eg affects the area of efgh?

- the area of efgh is $\frac{1}{4}$ of the area of the rectangle if e and g are not midpoints.
- the area of efgh is $\frac{1}{2}$ of the area of the rectangle only if e and g are midpoints.
- the area of efgh is always $\frac{1}{2}$ of the area of the rectangle.
- the area of efgh is always $\frac{1}{4}$ of the area of the rectangle.

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": [ "Area of a Kite", "Inscribed Polygons" ], "current_concepts": [ "Area of a Kite", "Inscribed Polygons", "Geometric Area Relations" ] } ## Express the area of the rectangle and the kite $$ \begin{aligned} &\text{Let the width of the rectangle be } w \text{ and the height be } h.\ &\text{Area of the rectangle: } A_{\text{rect}} = w \cdot h\ &\text{Since } EG \text{ is parallel to the horizontal sides, its length is } w.\ &\text{Since } F \text{ and } H \text{ are midpoints of the top and bottom sides, the vertical distance between them is } h.\ &\text{The diagonals of kite } EFGH \text{ are } EG \text{ and } FH, \text{ which are perpendicular.}\ &\text{Area of the kite: } A_{\text{kite}} = \frac{1}{2} \cdot EG \cdot FH = \frac{1}{2} \cdot w \cdot h \end{aligned} $$ ## Determine the relationship between the areas $$ \begin{aligned} &A_{\text{kite}} = \frac{1}{2} A_{\text{rect}}\ &\text{This relationship depends only on } EG = w \text{ and } FH = h.\ &\text{The vertical position of } EG \text{ (whether } E \text{ and } G \text{ are midpoints or not) does not affect the lengths of the diagonals.}\ &\text{Thus, the area of } EFGH \text{ is always } \frac{1}{2} \text{ of the area of the rectangle.} \end{aligned} $$

The area of EFGH is $\frac{1}{4}$ of the area of the rectangle if E and G are not midpoints.

The area of EFGH is $\frac{1}{2}$ of the area of the rectangle only if E and G are midpoints.

The area of EFGH is always $\frac{1}{2}$ of the area of the rectangle.

The area of EFGH is always $\frac{1}{4}$ of the area of the rectangle.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Area of a Kite" ] }

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Question

Explanation:

Express the area of the rectangle and the kite

Determine the relationship between the areas

Answer: