In the quadrilateral given below, AB || DC. E and F are mid-points of non-parallel sides AD and BC respectively. Calculate :

(i) EF if AB = 6 cm and DC = 4 cm

(ii) AB if DC = 8 cm and EF = 9 cm.

ABCD is a trapezium in which AB || DC and E, F are mid-points of AD and BC respectively.

Join CE and produce it to meet BA produced at G.

In △EDC and △EAG,

ED = EA (∵ E is mid-point of AD)

∠CED = ∠ GEA (Vertically opposite ∠s)

∠ECD = ∠EGA (Alternate ∠s)

∴ △EDC ≅ △EAG

⇒ CD = GA and EC = EG (c.p.c.t.)

In △CGB,

E is mid-point of CG

F is mid-point of BC

∴ By mid-point theorem, EF || AB and EF = $\dfrac{1}{2}$GB.

But GB = GA + AB = CD + AB

∴ EF = $\dfrac{1}{2}$(AB + CD) …..(1)

(i) Given,

AB = 6 cm and DC = 4 cm,

Putting these values in eq (1) we get,

EF = $\dfrac{1}{2}$(6 + 4)

= $\dfrac{1}{2}$ x 10

= 5 cm

Hence, EF = 5 cm.

(ii) Given,

DC = 8 cm and EF = 9 cm

Putting these values in eq (1) we get,

9 = $\dfrac{1}{2}$(AB + 8)

⇒ 18 = AB + 8

⇒ AB = 18 - 8 = 10 cm

Hence, AB = 10 cm.