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

Steps to Solve

1. Identify the quadrilateral and properties In this problem, you are dealing with quadrilateral ( ABCD ) where ( AB = 42 , \text{cm} ), ( BC = 21 , \text{cm} ), ( CD = 29 , \text{cm} ), ( DA = 34 , \text{cm} ), and the diagonal ( BD = 20 , \text{cm} ).

2. Use the formula for area For a quadrilateral given the lengths of its sides and one diagonal, you can use Brahmagupta's formula. The area ( A ) of quadrilateral ( ABCD ) can be calculated in two triangles ( ABD ) and ( BCD ):

   $$ A = \frac{1}{2} \times d \times (h_1 + h_2) $$ where ( d ) is the diagonal and ( h_1, h_2 ) are the heights of triangles ( ABD ) and ( BCD ).

3. Calculate the area of triangles using Heron’s Formula You will need to calculate the semi-perimeter ( s ) for each triangle:

   For ( \triangle ABD ):

   • Calculate the semi-perimeter: $$ s_1 = \frac{AB + AD + BD}{2} = \frac{42 + 34 + 20}{2} = 48 $$
   • Calculate the area using Heron’s formula: $$ A_1 = \sqrt{s_1(s_1 - AB)(s_1 - AD)(s_1 - BD)} = \sqrt{48(48 - 42)(48 - 34)(48 - 20)} $$

   For ( \triangle BCD ):

   • Calculate the semi-perimeter: $$ s_2 = \frac{BC + CD + BD}{2} = \frac{21 + 29 + 20}{2} = 35 $$
   • Calculate the area using Heron’s formula: $$ A_2 = \sqrt{s_2(s_2 - BC)(s_2 - CD)(s_2 - BD)} = \sqrt{35(35 - 21)(35 - 29)(35 - 20)} $$

4. Sum the areas of the two triangles Finally, add the area of the two triangles to get the total area of quadrilateral ( ABCD ):

   $$ A_{total} = A_1 + A_2 $$