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ABCD is a trapezium, AB is parallel to CD and CD is parallel to EF and AD and BC are non-parallel sides where AD = BC, then which statements (s) is/are correct? A) If E and F are midpoints of side AD and BC respectively, then the length of EF is half of the sum of parallel sides. B) If E and F are not midpoints of sides AD and BC, then EF = (AE × DC + FC × AB)/AD C) If E and F are midpoints of sides AD and BC respectively, then (ar ABFE) : (ar DCFE) = (3AB + DC) : (3DC + AB) 

ABCD is a trapezium, AB is parallel to CD and CD is parallel to EF and AD and BC are non-parallel sides where AD = BC, then which statements (s) is/are correct? A) If E and F are midpoints of side AD and BC respectively, then the length of EF is half of the sum of parallel sides. B) If E and F are not midpoints of sides AD and BC, then EF = (AE × DC + FC × AB)/AD C) If E and F are midpoints of sides AD and BC respectively, then (ar ABFE) : (ar DCFE) = (3AB + DC) : (3DC + AB)  Correct Answer A, B and C

Given,

ABCD is a trapezium,

AB || CD

CD || EF

AD = BC

Concept:

If △ABC ∼ △PQR, then

AB/PQ = BC/QR = AC/PR

Calculation:

Join A to C, then

In △ABC and ΔPFC

PF/AB = FC/BC

⇒ PF = FC/BC × AB

In △ADC and ΔAEP

EP/DC = AE/AD

⇒ EP = AE/AD × DC

EF = EP + PF

⇒ EF = (AE × DC)/AD + (FC × AB)/BC

⇒ EF = (AE × DC + FC × AB)/AD

Statement B is correct.

If E and F are midpoints, then

AE = ED = AD/2

AE = ED = BF = FC

EF = (AE × DC + AE × AB)/2AE

⇒ EF = /2AE

⇒ EF = (DC + AB)/2

Statement A is correct.

Let height of the trapezium be 2h,

If E and F are two midpoints, then

Height of trapezium ABFE = Height of trapezium DCFE = h cm

Area of trapezium ABFE = (1/2) × (AB + EF) × h

Area of trapezium ABFE = (1/2) × × h

Area of trapezium ABFE = (1/2) × × h

Area of trapezium ABFE = (1/2) × × h

Similarly,

Area of trapezium DCFE = (1/2) × (DC + EF) × h

Area of trapezium DCFE = (1/2) × × h

Area of trapezium DCFE = (1/2) × × h

Area of trapezium DCFE = (1/2) × × h

Area of ABFE : Area of DCFE = (1/2) × × h : (1/2) × × h = (3AB + DC) : (3DC + AB)

∴ All the statements are true. 

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