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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.

Q2.    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.

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From the figure:

quadrilateral area heron

We have joined the AC to form two triangles so that the calculation of the area will be easy.

The area of the triangle ABC can be calculated by Heron's formula:

So, the semi-perimeter, where a = 3 cm,\ b =4cm\ and\ c = 5cm.

s = \frac{a+b+c}{2} = \frac{3+4+5}{2} = 6cm

Heron's Formula for calculating the area:

A = \sqrt{s(s-a)(s-b)(s-c)}

= \sqrt{6(6-3)(6-4)(6-5)}= \sqrt{6(3)(2)(1)} = \sqrt{36} = 6\ cm^2

And the sides of the triangle ACD are a' =4cm,\ b' = 5cm\ and\ c' = 5cm.

So, the semi-perimeter of the triangle:

s' = \frac{a'+b'+c'}{2} = \frac{4+5+5}{2} = \frac{14}{2} = 7cm

Therefore, the area will be given by, Heron's formula

A = \sqrt{s'(s'-a')(s'-b')(s'-c')}.

= \sqrt{7(7-4)(7-5)(7-5)}

= \sqrt{7(3)(2)(2)} = 2\sqrt{21} = 9.2\ cm^2\ \ \ \ (Approx.)  

Then, the total area of the quadrilateral will be:

= Area\ of\ triangle\ ABC + Area\ of\ triangle\ ACD

\Rightarrow Total\ area\ of\ quadrilateral\ ABCD = 6+9.2 = 15.2\ cm^2

Hence, the area of the quadrilateral ABCD is 15.2 \ cm^2.

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