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

Explanation:

Step1: Recall Kite Area Formula

The area of a kite is given by $\frac{1}{2} \times d_1 \times d_2$, where $d_1$ and $d_2$ are the lengths of the diagonals.

Step2: Analyze Rectangle and Kite Diagonals

Let the length of the rectangle be $l$ (along $EG$) and the width be $w$ (perpendicular to $EG$). The diagonals of the kite: $EG$ has length $l$ (same as the rectangle's length), and the other diagonal (connecting $F$ and $H$) has length $w$ (same as the rectangle's width, since $F$ and $H$ are midpoints, so the distance between them is the width of the rectangle).

Step3: Calculate Kite and Rectangle Areas

Area of rectangle: $A_{rectangle} = l \times w$.
Area of kite: $A_{kite} = \frac{1}{2} \times l \times w$.
So, $A_{kite} = \frac{1}{2} A_{rectangle}$, regardless of the position of $EG$ (as long as $F$ and $H$ are midpoints, making the other diagonal equal to the rectangle's width). The first option says $\frac{1}{4}$, which is wrong. The third option's condition (E and G not midpoints) is irrelevant here as E and G are endpoints on the rectangle's sides, and the key is $F$ and $H$ being midpoints.

Answer:

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