Q:

Find the area of the triangles ABD and BCD using Heron’s formula. Hence find the area of quadrilateral ABCD.

Accepted Solution

A:
Answer:The Area of quadrilateral ABCD is 36 cm²  Step-by-step explanation:Given in the figure as :ABD and BCD is a triangle Length of sides of Δ ABD is:AD = 3 cmAB = 4 cmBD = x = [tex]\sqrt{(AB)^{2}+ (AD)^{2}}[/tex] Or, BD = [tex]\sqrt{(4)^{2}+ (3)^{2}}[/tex] = [tex]\sqrt{25}[/tex] = 5 cm Length of sides of ΔCBD is :BC = 13 cmCD = 12 cmNow By Heron's formulaArea of triangle ABD = [tex]\sqrt{s (s -a)(s-b) (s-c)}[/tex]And  s = [tex]\frac{AB + BD +DA}{2}[/tex] Or,    s = [tex]\frac{4 + 5 +3}{2}[/tex] Or,    s = 6 cm∴ Area of triangle ABD = [tex]\sqrt{6 (6 -4)(6-5) (6-3)}[/tex]Or, Area of triangle ABD = [tex]\sqrt{36}[/tex] = 6 cm²           Similarly  The area of  Triangle CBD =  [tex]\sqrt{s (s -a)(s-b) (s-c)}[/tex]     And  s = [tex]\frac{CB + BD +DC}{2}[/tex] Or,    s = [tex]\frac{13 + 5 +12}{2}[/tex] Or,    s = 15 cm ∴ Area of triangle CBD = [tex]\sqrt{15 (15 -13)(15-5) (15-12)}[/tex]Or, Area of triangle CBD = [tex]\sqrt{900}[/tex] = 30 cm²      The Area of quadrilateral ABCD = Area Δ ABD + Area Δ CBD       Or,The Area of quadrilateral ABCD =  6 cm²   +  30 cm²  = 36 cm²        Hence The Area of quadrilateral ABCD is 36 cm²    Answer