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.

Explanation:

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<reasoning>

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

\[

$$\begin{aligned} &A_{\text{kite}} = \frac{1}{2} A_{\text{rect}}\\ &\text{This relationship depends only on } EG = w \text{ and } FH = h.\\ &\text{The vertical position of } EG \text{ (whether } E \text{ and } G \text{ are midpoints or not) does not affect the lengths of the diagonals.}\\ &\text{Thus, the area of } EFGH \text{ is always } \frac{1}{2} \text{ of the area of the rectangle.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>The area of EFGH is \(\frac{1}{4}\) of the area of the rectangle if E and G are not midpoints.</mcq-option>
<mcq-option>The area of EFGH is \(\frac{1}{2}\) of the area of the rectangle only if E and G are midpoints.</mcq-option>
<mcq-correct>The area of EFGH is always \(\frac{1}{2}\) of the area of the rectangle.</mcq-correct>
<mcq-option>The area of EFGH is always \(\frac{1}{4}\) of the area of the rectangle.</mcq-option>
</answer>

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Answer:

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<reasoning>

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

\[

$$\begin{aligned} &A_{\text{kite}} = \frac{1}{2} A_{\text{rect}}\\ &\text{This relationship depends only on } EG = w \text{ and } FH = h.\\ &\text{The vertical position of } EG \text{ (whether } E \text{ and } G \text{ are midpoints or not) does not affect the lengths of the diagonals.}\\ &\text{Thus, the area of } EFGH \text{ is always } \frac{1}{2} \text{ of the area of the rectangle.} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>The area of EFGH is \(\frac{1}{4}\) of the area of the rectangle if E and G are not midpoints.</mcq-option>
<mcq-option>The area of EFGH is \(\frac{1}{2}\) of the area of the rectangle only if E and G are midpoints.</mcq-option>
<mcq-correct>The area of EFGH is always \(\frac{1}{2}\) of the area of the rectangle.</mcq-correct>
<mcq-option>The area of EFGH is always \(\frac{1}{4}\) of the area of the rectangle.</mcq-option>
</answer>

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