Question

In the given figure, ABCD is a quadrilateral such that AB : BC : CD = 3 : 2 : 4. If the area of the quadrilateral is 448 $c{m}^2$, then the perimeter of the quadrilateral is

• A. 80 cm
• B. $8[9+2\sqrt{5}]$ cm
• C. $8[9+\sqrt{5}]$ cm
• D. $9[8+\sqrt{5}]$ cm

Answer 1

The correct option is C $8[9+\sqrt{5}]$ cm



Given: AB : BC : CD = 3 : 2 : 4
Let AB = 3K, BC = 2K and DC = 4K, where K is any natural number.
Drawing AE ⊥ DC.
Now, DE = DC – EC
= 4K – 3K ($\because$ AB = EC)
= K
$\because$Area of quadrilateral ABCD = Area of rectangle ABCE + Area of $\Delta$ADE
$=AB\times BC+\frac{1}{2}\times DE\times AE$
$=3K\times 2K+\frac{1}{2}\times K\times 2K$ ( AE = BC = 2K)
$=6K^2+K^2$
$=7K^2$

$\Rightarrow 7K^2=448$
$\Rightarrow K^2=64$
$\Rightarrow K=8$
$\therefore$ AB = 24 cm, BC = AE =16 cm, DC = 32 cm and DE = 8 cm In $\Delta ADE$, by Pythagoras theorem,
$(AD{)}^2=(AE{)}^2+(DE{)}^2$
$\Rightarrow (AD{)}^2=(16{)}^2+(8{)}^2$
$\Rightarrow AD=8\sqrt{5}$
$\therefore$ Perimeter of a quadrilateral ABCD = AB + BC+CD +AD
$=24+16+32+8\sqrt{5}$
$=72+8\sqrt{5}$
$=8(9+\sqrt{5})$cm
Hence, the correct answer is option (c).