Diagonals of the rhombus

How long are the diagonals e, f in the diamond, if its side is 5 cm long and its area is 20 cm²?

Correct answer:

e =  4.4721 cm
f =  8.9443 cm

Step-by-step explanation:

$$\begin{gathered} a=5\text{ cm } \\ S=20{\text{cm}}^2 \\ S=ah \\ h=S\mathrm{/}a=20\mathrm{/}5=4\text{ cm } \\ h=a\cdot \sin (A) \\ A={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arcsin (h\mathrm{/}a)={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arcsin (4\mathrm{/}5)\doteq 53.1301^{\circ } \\ e=a\cdot \sqrt{2-2\cdot \cos A^{\circ }}=a\cdot \sqrt{2-2\cdot \cos 53.1301023542^{\circ }}=5\cdot \sqrt{2-2\cdot \cos 53.1301023542^{\circ }}=5\cdot \sqrt{2-2\cdot 0.6}=4.472=4.4721\text{ cm} \end{gathered}$$

$f=a\cdot \sqrt{2+2\cdot \cos A^{\circ }}=a\cdot \sqrt{2+2\cdot \cos 53.1301023542^{\circ }}=5\cdot \sqrt{2+2\cdot \cos 53.1301023542^{\circ }}=5\cdot \sqrt{2+2\cdot 0.6}=8.944=8.9443\text{ cm}$

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