Rhombus - diagonals

Rhombus ABCD has side a = 80 cm and side b = 50 cm. Diagonals u1 and u2 make an angle of 60 degrees with each other.
Calculate the area of ​​the rhombus.

Correct answer:

S =  3900 cm²

Step-by-step explanation:

$$\begin{gathered} a=80\text{cm} \\ b=50\text{cm} \\ \Phi =60{}^{\circ } \\ \phi =180-\Phi =180-60=120{}^{\circ } \\ {u}_1=2\cdot x \\ {u}_2=2\cdot y \\ {a}^2=x^2+y^2-2xy\cdot \cos \Phi \\ {b}^2=x^2+y^2-2xy\cdot \cos \phi \\ {a}^2-b^2=2xy(\cos \Phi -\cos \phi ) \\ z=xy \\ z=\frac{a^2-b^2}{2\cdot (\cos \Phi -\cos \phi )}=\frac{a^2-b^2}{2\cdot (\cos 60°-\cos 120°)}=\frac{80^2-50^2}{2\cdot (\cos 60°-\cos 120°)}=\frac{80^2-50^2}{2\cdot (0.5-(-0.5))}=1950 \\ S=\frac{1}{2}{u}_1\cdot u_2=2\cdot x\cdot y \\ S=2\cdot z=2\cdot 1950=3900{\text{cm}}^2 \end{gathered}$$

Try another example