12) In a rhombus with an area of $80 \mathrm{~cm}^{2}$, one diagonal is twice as long as the other. How long is the side of the rhombus?
(A) $8 \mathrm{~cm}$
(B) $\sqrt{80} \mathrm{~cm}$
(C) $10 \mathrm{~cm}$
(D) $20 \mathrm{~cm}$
(E) cannot be determined.
12) The answer is $\mathbf{( C )}$.

Let $d$ be the shorter diagonal of the rhombus and $D=2 d$ be the longer diagonal. The area $S$ of the rhombus is given by

$$
S=\frac{D \cdot d}{2}=\frac{2 d \cdot d}{2}=d^{2}
$$

thus $d=\sqrt{S}=\sqrt{80} \mathrm{~cm}$ and $D=2 \sqrt{80} \mathrm{~cm}$.

The Pythagorean theorem allows us to calculate the length of the side $l$ :

$$
l=\sqrt{\left(\frac{d}{2}\right)^{2}+\left(\frac{D}{2}\right)^{2}}=\sqrt{\frac{80}{4}+\frac{320}{4}}=\sqrt{20+80}=10 \mathrm{~cm}
$$