1. 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?
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By: anil on 05 May 2019 02.16 am
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 $$ riangle$$ AOB, $$sin (angle OAB) = frac{OB}{AB}$$ => $$sin(45) = frac{OB}{6}$$ => $$frac{1}{sqrt{2}} = frac{OB}{6}$$ => $$OB = frac{6}{sqrt{2}} = 3sqrt{2}$$ cm Similarly, OA = $$3sqrt{2}$$ cm In $$ riangle$$ BCO, $$tan(angle OCB) = frac{OB}{OC}$$ => $$tan(30) = frac{3sqrt{2}}{OC}$$ => $$frac{1}{sqrt{3}} = frac{3sqrt{2}}{OC}$$ => $$OC = 3sqrt{6}$$ cm $$ herefore$$ AC = OA + OC = $$3sqrt{2}+3sqrt{6}$$ = $$3(sqrt{2}+sqrt{6})$$ cm => Ans - (D)
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