Question

If E, F, G and H are respectively the midpoints of the sides of a ❑ᵐPQRS, show that EFGH = 1/2 (PQRS).

Answer 1

Step-by-step explanation:

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Answer 2

Answer:

We have given that ABCD is a parallelogram and E, F, G, H are the midpoints of the sides of PQ, QR, RS, and SP respectively.

We have to prove:

EFGH = 1/2(PQRS)

Proof:

Since F and H are the midpoints of QR and SP respectively,

PH = 1/2PS and QF = 1/2QR

PQRS is a parallelogram

QR = PS and QR  ║PS

$\dfrac 12PS = \dfrac 12QR$

and QR  ║PS

PH = QF.

PQFH is a parallelogram.

Since parallelogram FHPQ and FHE are on the same base and between the same parallels HF and PQ,

$ar (\triangle FHE) = \dfrac 12 ar(FHPQ),$

Similarly,

$ar(\triangle FHG) = \dfrac 12 ar(FHSR)$

Now,

$ar (\triangle FHE) +ar (\triangle FHG) = \dfrac 12ar(FHPQ) + \dfrac12ar(FHSR)\\\\ar(EFGH) =\dfrac 12(ar(FHPQ) + ar(FHSR))\\\\ar(EFGH) =\dfrac 12 ar(PQRS)$

Hence Proved.