Question

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 1

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