Question

the diagonal bd of a rectangle abcd is at an angle of 30° with the side bc. Calculate the area of the triangle, if bd=4 units

Answer 1

Explanation:

ABCD is a rectangle and BD is diagonal such that:

`\angle DBC = 30\degree</p><p> (given) \\\\</p><p>\angle BCD = 90\degree</p><p>(\angle\:of\:a\:rectangle) \\\\</p><p> \therefore \angle CDB= 60\degree </p><p> (remaining\:\angle\:of \:\triangle)</p><p>\\\\</p><p> \therefore \triangle BCD\:is \:a\:30\degree , \: 60\degree \:\&\:90\degree \:\triangle. \\\\</p><p> \therefore CD = (1)/(2) * BD (side\:oppossite \:to \:30\degree)\\\\</p><p> \therefore CD = (1)/(2) * 4\\\\</p><p> \therefore CD =2\: units\\\\</p><p>\&\\\\</p><p> BC = (\sqrt 3)/(2) * BD(side\:oppossite \:to \:60\degree)\\\\</p><p> \therefore BC = (\sqrt 3)/(2) * 4\\\\</p><p>\therefore BC =2{\sqrt 3}\: units\\\\</p><p> Now\\\\</p><p>A(\triangle BCD) = (1)/(2) * BC* CD\\\\</p><p> \therefore A(\triangle BCD) = \frac {1}{2} * {2\sqrt 3} * 2\\\\</p><p>\red{\boxed {\therefore A(\triangle BCD) ={2\sqrt 3}\: units^2}}`