Area of  Trapezium

Area of trapezium ABCD = Area of ∆ ABD + Area of ∆ CBD

= 1/2 × a × h + 1/2 × b × h

= 1/2 × h × (a + b)

= 1/2 (sum of parallel sides) × (perpendicular distance between them)

2. ABCD is a trapezium in which AB ∥ CD, AD ⊥ DC, AB = 20 cm, BC = 13 cm and DC = 25 cm. Find the area of the trapezium. 

Therefore, AB = DP = 20 cm

So, PC = DC – DP

= (25 – 20) cm

= 5 cm

Now, area of trapezium ABCD = Area of rectangle ABPD + Area of △ BPC

△BPC is right angled at ∠BPC

Therefore, using Pythagoras theorem,

      BC² = BP² + PC²

     13² = BP² + 5²

⇒ 169 = BP² + 25

⇒ 169 – 25 = BP²

⇒ 144 = BP²

⇒ BP = 12

Now, area of trapezium ABCD = Area of rectangle ABPD + Area of ∆BPC

                                                = AB × BP + 1/2 × PC × BP 

                                                = 20 × 12 + 1/2 × 5 × 12 

                                                = 240 + 30 

                                                = 270 cm²

3. The area of a trapezium is 165 cm² and its height is 10 cm. If one of the parallel sides is double of the other, find the two parallel sides.

Solution:

Let one side of trapezium is x, then other side parallel to it = 2x

Area of trapezium = 165 cm²

Height of trapezium = 10 cm

Now, area of trapezium = 1/2 (p₁ + p₂) × h

⇒ 165 = 1/2(x₁ + 2x) × 10

⇒ 165 = 3x × 5

⇒ 165 = 15x

⇒ x = 165/15

⇒ x = 11

Therefore, 2x = 2 × 11 = 22

Therefore, the two parallel sides are of length 11 cm and 22 cm.

4. Find the area of a trapezium whose parallel sides are AB = 12 cm, CD = 36 cm, and the non-parallel sides are BC = 15 cm and AG = 15 cm.

Now CE = 15 cm

Since, DC = 12 cm so, AE = 12 cm

Also, EB = AB – AE = 36 – 12 = 24 cm

Now, in ∆ EBC

S = (15 + 15 + 24)/2

= 54/2

= 27

= √(27 × 12 × 12 × 3)

= √(3 × 3 × 3 × 3 × 2 × 2 × 2 × 2 × 3 × 3)

= 3 × 3 × 3 × 2 × 2

= 108 cm²

Draw CP ⊥ EB.

Area of ∆EBC = 1/2 × EB × CP

108 = 1/2 × 24 × CP

108/12 = CP

⇒ CP = 9 cm Therefore, h = 9 cm

Now, area of triangle = √(s(s – a) (s – b) (s – c))

= √(27 (27 – 15) (27 – 15 ) (27 – 24))

Now, area of trapezium = 1/2(p₁ + p₂) × h

= 1/2 × 48 × 9

= 216 cm²