Question

ABCD is a rectangle. the points E, F, G and H are chosen on the sides AB, BC, CD and DA respectively so EFGH is a rectangle. Furthermore CF = AH = 9m, DG = 15m. If AC is parallel to GH, show that ABCD is a square and find its side length. What is the side length???

Answer 1

Step-by-step explanation:

Given that :- ABCD is a rectangle, E,F,G,H are m.p of AB, BC, CD, DA, Ar (EFGH) = 16.

To Find :- Ar (ABCD).

Construction:- Join FH.

Proof:-

Now,

In quad ABFH,

Ar (EFH) = 1/2 Ar (ABFH). ..................1).

Similarly in quad, DCFH,

Ar ( GFH ) = 1 /2 Ar (DCFH). ..............2).

From 1) and 2), we get:-

Ar (EFGH) = 1/2 Ar(ABCD). ...............3).

Therefore Ar ( EFGH ) = 16 cm^2 (given).

From 3, Ar (ABCD) = 2 x 16 = 32 cm^2

see since area of the quadrilateral ( parallelogram) is 16cm² which is the area formed by the midpoints of the rectangle so the area of the rectangle must be equal to twice the ar( EFGH) . and so the area of the rectangle is 32cm².