Question

ABCD is a rectangle and lies DX, DY and XY are drawn as shown Area of $\Delta AXD$ is 5, area of $\Delta BXY$ is 4 and area of $\Delta CYD$ is 3 If the area of $\Delta DXY$ can be expressed as $\sqrt{x}$ where $xϵN$, then x is equal to

• A. $72$
• B. $75$
• C. $84$
• D. $96$

Answer 1

The correct option is C $84$

The figure is attached in as image
Fromt the figure:
Area of $△AXD=5$
$\frac{1}{2}bs=5\Rightarrow bs=10$
Area of DCY=3
$\frac{1}{2}lr=3\Rightarrow lr=6$
Area of BXY=4
$\frac{1}{2}(l-s)(b-r)=3\Rightarrow ab-bs-lr+rs=8$
$lb-10-6+rs=8$
$lb+rs=24$
We already have $lr=6$ and $bs=10$
$lbrs=60$
$lb+rs=24$
Solve both the above equations
$lb-rs=\sqrt{(lb+rs{)}^2-4lbrs}=\sqrt{{24}^2-4\left(60\right)}=\sqrt{336}$
$lb=\frac{1}{2}(24+\sqrt{336})=12+\sqrt{84}$
Area of DXY = Area of Rectangle - Area of other 3 triangles
Area of DXY= $12+\sqrt{84}-5-4-3=\sqrt{84}$
So $x=84$