The shorter diagonal of a rhombus makes an angle of 78 degrees with a side of the rhombus. If the length of the shorter diagonal is 24, find to the nearest tenth the length of: (a) a side of the rhombus and (b) the longer diagonal.
The shorter diagonal of a rhombus makes an angle of 78 degrees with a side of the rhombus. If the length of the shorter diagonal is 24, find to the nearest tenth the length of: (a) a side of the rhombus and (b) the longer diagonal.
As a rhombus has all 4 sides of equal length each must be 24. From the angle of 78 it's supplementary angle must be 180-78 = 112 degrees.
From there the cos rule can be used. Let c be the diagonal across the rhombus such that it meets the corners at the 78 degree angles:
$\displaystyle c^2 = a^2+b^2-2abcosC$
$\displaystyle c^2 = 2a^2 - 2a^2cosC = 2a^2(1-cosC)$
a = 24 and C = 102