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State true or false

$A B C D$ is a parallelogram. $E$ is a parallelogram. is a point on $B A$ is a point on such that $B E = 2 E A$ such that and $F$ and is a point on $D C$ is a point on , such that $D F = 2 F C$ , such that . Then $A E C F$ . Then is a parallelogram whose area is one third of the area of parallelogram $A B C D$ is a parallelogram whose area is one third of the area of parallelogram .

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True

False

answer is A.

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Detailed Solution

Given that,

A B C D

is a parallelogram.

E

is a point on

B A

such that

B E = 2 E A

and

F

is a point on

D C

, such that

D F = 2 F C

.
Draw a parallelogram,
Question Image

B E = 2 E A ⇒ A B − A E = 2 A E ⇒ A B = 3 A E

Similarly,

D F = 2 F C ⇒ D C − F C = 2 F C ⇒ D C = 3 F C

Since,

A B = D C

. [opposite sides of a parallelogram are equal]
Then,

A E = F C

.
So,

A E ‖ F C

such that

A E = F C

.
Then,

A E C F

is a parallelogram.
Clearly, parallelogram ABCD and AEFC have the same altitude (suppose h) and

A E = 13 A B

.
Area of parallelogram

(A B C D) = (A B × h)

a r (∥ g m A E C F) = h × A E = 13 × h × A B = 13 ​ a r (∥ g m A B C D)

Hence, area of parallelogram

A E C F = 13

area of parallelogram

A B C D

.
Hence, the statement is true.
Therefore, option 1 is correct.

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