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 statements describes how the location of segment EG affects the area of EFGH? A.) the area of EFGH is 1/4 of the area of the rectangle if E and G are not midpoints B.) The area of EFGH is 1/2 of the area of the rectangle only if E and G are midpoints C.) The area of EFGH is always 1/2 of the area of the rectangle. D.) The area of EFGH is always 1/4 of the area of the rectangle.

Answer 1

C.) The area of EFGH is always ¹/₂ of the area of the rectangle.

Explanation:

If EG is parallel to the side of the rectangle then lenght of EG is equal to width of rectangle.

If F and H are midpoints of sides of rectangle then FH is parallel to the side of rectangle {wich is perpendicular to the side parallel to EG}. That means the lenght of FH is equal to lenght of rectangle, and FH is perpendicular to EG.

Then FH is sum of hights of triangles EFG and EHG
`(FH=H_{_(\Delta EFG)}+H_{_(\Delta EHG)})`, and the area of EFGH is sum of areas of triangles EFG and EHG
`(A_(kite)=P_{_(\Delta EFG)}+P_{_(\Delta EHG)})`.

So the area of the rectangle:
`\bold{A_(rectangle)=EG\cdot FH}`

The area of the kite:

`A_(kite)=P_{_(\Delta EFG)}+P_{_(\Delta EHG)}\\\\A_(kite)=\frac12 EG\cdot H_{_(\Delta EFG)}+\frac12 EG\cdot H_{_(\Delta EHG)}\\\\ A_(kite)=\frac12 EG\cdot (H_{_(\Delta EFG)}+H_{_(\Delta EHG)})\\\\A_(kite)=\frac12 EG\cdot FH\\\\A_(kite)=\frac12 A_(rectangle)`

No matter the height of the triangles, so no matter the location of the EG