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If E and F are the mid-points of the diagonals AC and BD in trapezium ABCD. Then, EF can be written as which of the following?

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12 A B

12 C D

12 (A B + C D)

12 (A B − C D)

answer is D.

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Detailed Solution

ABCD is the given trapezium and we are also given that E, F are mid-points of diagonal AC and BD respectively.
Let us consider the given figure. Draw a line CG such that F is the midpoint of CG.
Question Image

.
In the trapezium, AB||CD.
In

Δ C D F

and

Δ G B F

D F = B F

… [F is the midpoint of diagonal BD]

∠ D C F = ∠ B G F

… [ DC||GB and CG are transversals]

∠ C D F = ∠ G B F

. … [ DC||GB and BD are transversals]
Hence,

Δ C D F ≅ Δ G B F

. [By A.S.A congruence theorem]

C D = G B

[By C.P.C.T] …… (i)
Consider the

Δ C A G

, points E and F are the mid-points of AC and CG respectively. By mid-point theorem, we can write that,

E F = 12 A G E F = 12 A B − G B

Using equation (i),

E F = 12 A B − C D

.
Hence, option 4 is the correct answer.

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