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:

Step1: Recall the formula for the area of a kite

The area of a kite \(A=\frac{1}{2}d_1d_2\), where \(d_1\) and \(d_2\) are the diagonals of the kite. In this case, let the length of the rectangle be \(l\) and the width be \(w\). The diagonal \(EG\) of the kite \(EFGH\) is parallel to the side of the rectangle. Let \(EG = x\) (parallel to the length of the rectangle) and the other diagonal (perpendicular to \(EG\)) be \(y\).
We can also use the fact that the area of the rectangle \(A_{r}=l\times w\).
If we consider the two - triangle decomposition of the kite. The area of the kite \(A_{k}\) can be thought of as the sum of the areas of two triangles with the same base \(EG\).
Let the length of \(EG\) be \(b\) and the sum of the heights of the two triangles (with base \(EG\)) be \(h\) (which is equal to the width of the rectangle).
The area of a triangle is \(A_{\triangle}=\frac{1}{2}bh\). The area of the kite \(A_{k}=\frac{1}{2}EG\times\) (sum of the heights of two triangles with base \(EG\))
Since the sum of the heights of the two triangles with base \(EG\) is equal to the width of the rectangle and \(EG\) is equal to the length of the rectangle (by the property of the inscribed kite in the rectangle)
\(A_{k}=\frac{1}{2}\times\) (length of the rectangle)\(\times\) (width of the rectangle)

Step2: Compare with the area of the rectangle

The area of the rectangle \(A_{r}=\text{length}\times\text{width}\)
We know that \(A_{k}=\frac{1}{2}A_{r}\) regardless of the position of \(E\) and \(G\) (as long as \(EG\) is parallel to the side of the rectangle and \(F\) and \(H\) are mid - points). The formula for the area of the kite (sum of two triangles \(\frac{1}{2}EG\times h_1+\frac{1}{2}EG\times h_2=\frac{1}{2}EG\times(h_1 + h_2)\), where \(h_1+h_2\) is the width of the rectangle and \(EG\) is the length of the rectangle) does not depend on \(E\) and \(G\) being mid - points for the ratio \(\frac{A_{k}}{A_{r}}\)

Answer:

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