Question

Find the area of quad. ABCD in which AB = 42 cm, BC = 21 cm, CD =  29 cm, DA = 34 cm and diag. BD = 20 cm.

Answer 1

Given: AB = 42 cm, BC = 21 cm, CD = 29 cm, DA = 34 cm, BD = 20 cm. Split quad ABCD along diagonal BD into triangles ABD and BCD.

Step-wise calculation:

1. Triangle ABD (sides 42, 34, 20):

`s_1 = (42 + 34 + 20)/2`

= 48

Area₁ = `sqrt(s_1(s_1 - 42)(s_1 - 34)(s_1 - 20))`

= `sqrt(48·6·14·28)`

Compute product: 48·6·14·28 

= 112896 

= (8·42)² 

So Area₁ = 336 cm².

2. Triangle BCD (sides 21, 29, 20):

`s_2 = (21 + 29 + 20)/2`

= 35

Area² = `sqrt(s_2(s_2 - 21)(s_2 - 29)(s_2 - 20))`

= `sqrt(35·14·6·5)`

Compute product: 35·14·6·15

= 44100 

= 210²

So Area₂ = 210 cm².

3. Total area of quadrilateral ABCD = Area₁ + Area₂

= 336 + 210

= 546 cm²

Area(ABCD) = 546 cm².