ABCD is a kite where m angle A is 90° and m angleC is 60°. If length of AB is 6 cm, what is the length of diagonal AC?

Given : ABCD is a kite with AB = 6 cm 

$$\angle$$ DAB = 90° and $$\angle$$ BCD = 60°

To find : AC = ?

Solution : Diagonals of a kite intersect at 90° and bisect the angles opposite to them.

=> $$\angle$$ OAB = 45° and $$\angle$$ OCB = 30°

In $$\triangle$$ AOB, $$sin (\angle OAB) = \frac{OB}{AB}$$

=> $$sin(45) = \frac{OB}{6}$$

=> $$\frac{1}{\sqrt{2}} = \frac{OB}{6}$$

=> $$OB = \frac{6}{\sqrt{2}} = 3\sqrt{2}$$ cm

Similarly, OA = $$3\sqrt{2}$$ cm

In $$\triangle$$ BCO, $$tan(\angle OCB) = \frac{OB}{OC}$$

=> $$tan(30) = \frac{3\sqrt{2}}{OC}$$

=> $$\frac{1}{\sqrt{3}} = \frac{3\sqrt{2}}{OC}$$

=> $$OC = 3\sqrt{6}$$ cm

$$\therefore$$ AC = OA + OC = $$3\sqrt{2}+3\sqrt{6}$$

= $$3(\sqrt{2}+\sqrt{6})$$ cm

=> Ans - (D)