Question

The lengths of four sides and a diagonal of the given quadrilateral are indicated in the diagram. If $A$ denotes the area and $l$ the length of the other diagonal, then $A$ and $l$ are respectively

• A. $12\sqrt{6},4\sqrt{6}$
• B. $12\sqrt{6},5\sqrt{6}$
• C. $6\sqrt{6},4\sqrt{6}$
• D. $6\sqrt{6},5\sqrt{6}$

Answer 1

The correct option is A $12\sqrt{6},4\sqrt{6}$

For $\Delta ABC$, a $=$ 6 cm, b $=$ 5, c $=$ 7 cm

$\therefore s=\frac{6+5+7}{2}=9$ cm

$\therefore$ Area of $\Delta ABC=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{9\times (9-6)(9-5)(9-7)}$

$=\sqrt{9\times 3\times 4\times 2}$

$=3\times 2\sqrt{6}=6\sqrt{6}$

Thus, area of quadrilateral $=2\times$ Area of $\Delta ABC$
$=12\sqrt{6}$ sq cm

From congruency of triangles, it can be proved that $AE=ED=\frac{1}{2}AD$, and $AE\perp BC$

$AE=\frac{1}{2}AD=\frac{l}{2}$

Area of $\Delta ABC=\frac{1}{2}\times BC\times AE$

or $6\sqrt{6}=\frac{1}{2}\times 6\times \frac{l}{2}$

or $l=4\sqrt{6}$