The figure (i) given below is a trapezium. Find the length of BC and the area of the trapezium. Assume AB = 5 cm, AD = 4 cm, CD = 8 cm.

(a) Construct BN perpendicular to CD.

So, BADN is a rectangle.

As opposite sides of rectangle are equal.

∴ BN = AD = 4 cm and ND = BA = 5 cm.

From figure,

CN = CD – ND = 8 - 5 = 3 cm.

In right angle triangle BCN,

Using Pythagoras theorem,

⇒ BC² = BN² + CN²

⇒ BC² = 4² + 3²

⇒ BC² = 16 + 9 = 25

⇒ BC = $\sqrt{25}$ = 5 cm.

By formula,

Area of trapezium = $\dfrac{1}{2}$ × sum of parallel sides × height

= $\dfrac{1}{2}$ × (AB + CD) × AD

= $\dfrac{1}{2}$ × (5 + 8) × 4

= 13 × 2 = 26 cm².

Hence, BC = 5 cm and area of trapezium = 26 cm².