Question

ABCD is a parallelogram. W, X, Y and Z are points on sides AB, BC, CD, and DA respectively, such that A W = D Y . If ar ( A B C D ) = 400 c m 2 , what is the area of quadrilateral WXYZ?

Answer 1

Answer:

The area of WXYZ is 200 cm².

Step-by-step explanation:

[Refer to the attached figure.]

Construction

Join ZX and WY.

proof

Now consider the parallelogram AWOZ.

ZW is the diagonal of AWOZ.

We know that the diagonal of a parallelogram divides it into equal parts so,

          ar(AZW) = ar(OZW)  ---------(1)

Similarly, you can prove that

• ar(OXW) = ar(BXW) ----------(2)

• ar(OYZ) = ar(DYZ) ------------(3)

• ar(OYX) = ar(CYX) ------------(4)

NOW ar(ABCD) = ar(AZW) +ar(BXW) +ar(DYZ) +ar(CYX) +ar(OZW)+ar(OXW)+ar(OYZ)+ar(OYX).

=> ar(ABCD) = 2[ar(OZW)+ar(OXW)+ar(OYZ)+ar(OYX)]

=> ar(ABCD) = 2[ar(WXYZ)]

=> ar(WXYZ) = [ar(ABCD)]/2

=> ar(WXYZ) = 400/2=200 cm²