Question

find the area of the shaded region in the figure

Answer 1

you can see here,
∆ADC is a right angle ∆.
so,
use Pythagoras theorem , for finding AC.
AC² = AD² + CD²
AC² = 12² +5² = 169
AC = 13 cm

now,
area of shaded region = area of ∆ABC - area of ∆ADC


area of ∆ABC = √{s(s-a)(s-b)(s-c) }
where,
a = 13
b=14
c=15
S = (a + b+ c)/2 = 21

area of ∆ABC = √{21 ×8 × 7 × 6} =7 ×3 ×4 = 84 cm²

and area of ∆ADC = 1/2 × base × height
= 1/2 × 12 × 5 = 30 cm²

now,
area of shaded region = (84 - 30) cm² =54 cm²

Answer 2

area of abd = 30
ab = 13 (pythagorus theorem)
area of acb = by herons formula
area of adbc = area of abc - abd