Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, C

1.	Find the area of a quadrilateral ABCD in which AB = 3 cm, BC = 4 cm, CD = 4 cm, DA = 5 cm and AC = 5 cm.Chapter :- Herons formula
Answer» Given :- • A quadilateral ABCD • AB = 3 CM • BC = 4 cm • CD = 4 cm • AC = 5 cm • DA = 5 cm To Find :- • AREA of Quadilateral Concept :- • We will find the area of both triangles i.e Triangle ABC and Triangle ADC by herons formula • Then we will add both triangles area to get the area of a quadilateral Solution:- In triangle ABC $\sf\bold{\red{⟹ a = 5 \: cm}}$ $\sf\bold{\red{⟹ b = 4 \: cm}}$ $\sf\bold{\red{⟹ c = 3 \: cm}}$ $\sf \: \: ∴ \: Semi \: perimeter = \sf\bold{\red{\frac{a + b + c}{2} }}$ $⟹ \frac{5 + 4 + 3}{2}$ $⟹ \frac{12}{2}$ $⟹ {\sf {{\dfrac{{\cancel{{12}}^{6}}}{{\cancel{{2}}^{1}}}}}}$ $⟹6$ Therefore Semi perimeter = 6 cm Now area of Triangle ABC by herons formula $\: \sf\bold{\red{\sqrt{s(s - a)(s - b)(s - c)}}}$ we're S is semi perimeter which is equal to 6 cm a , b and c are THREE sides of the triangle which are 5 cm , 4 cm and 3cm respectively Putting the values $⟹ \sqrt{6(6 - 5)(6 - 4)(6 - 3)}$ $⟹ \sqrt{6 \times 1 \times 2 \times 3}$ $⟹ \sqrt{6 \times 2 \times 3}$ $⟹ \sqrt{36}$ $⟹ \sf \sqrt{6 {cm}^{2} }$ Now In triangle ADC $⟹ \sf\bold{\pink{a = 5\: cm}}$ $⟹ \sf\bold{\pink{b = 5\: cm}}$ $⟹ \sf\bold{\pink{c = \: 4 \: cm}}$ $\sf \: \: ∴ \: Semi \: perimeter = \sf\bold{\pink{\frac{a + b + c}{2} }}$ $⟹ \frac{5 + 5+ 4}{2}$ $⟹ \frac{14}{2}$ $⟹ {\sf { {\dfrac{{\cancel{{14}}^{7}}}{{\cancel{{2}}^{1}}}}}}$ $⟹7$ Therefore Semi perimeter = 7 cm Now area of Triangle ADC by herons formula $\sf\bold{\pink{\sqrt{s(s - a)(s - b)(s - c)}}}$ we're S is semi perimeter which is equal to 7 cm a , b and c are three sides of the triangle which are 5 cm , 5 cm and 4 cm respectively Putting the values $⟹ \sqrt{7(7 - 5)(7 - 5)(7 - 4)}$ $⟹ \sqrt{7 \times 2 \times 2 \times 3}$ $⟹ \sqrt{84}$ $⟹ \sf \: 9.165 \: {cm}^{2}$ Therefore , Area of Quadilateral ABCD = Area of Triangle ABC + Area of Triangle ADC = 6 + 9.165 $\sf= 15.165 {cm}^{2}$ Formula used :- $• \: \sf \: \: Semi \: perimeter = \sf\bold{\green{\frac{a + b + c}{2} }}$ • Herons formula , $⟹\sf\bold{\green {\sqrt{s(s - a)(s - b)(s - c)}}}$ Note :- ⟹ Figure in attachment