Question

The longer diagonal of a parallelogram measures 62 cm and makes an angle of 30 degrees with the base. Find the area of the parallelogram if the diagonals intersect at angle of 70 degrees.

Answer 1

The four small triangles determined by the diagonals of a parallelogram have equal areas.
So it is enough to find the area of one and multiply it by 4.
Let us find the area of a triangle having one side a and 30 snd 70 angles that have side a.
Let x be the the other side adjacent to 30 then
\[
\frac {x}{\sin 70}=\frac {a}{\sin 80}\\
x= {\sin 70}\frac {a}{\sin 80}=0.954189 a

\]
Area of the triangle
\[
\frac 12 x a \sin(30)=\frac 14 x a=\frac 1 4 0.954189 a^2=\frac 1 4 0.954189 \times31^2=229.244
\]
Area of the paralleogram
\[
4 \times 229.244=916.976
\]