<s>[INST] Given diagonals in a parallelogram. Find sides. Given a parallelogram with $d_1 = AC = 26$ cm, $d_2 = BD = 18$ cm and $\sin \displaystyle \angle AOD = \frac{12}{13}$.
Find $AB = a$ and $AD = b$.

What I did in order to solve it

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*Using the formula for the area

$S = \frac{d_1d_2\sin \displaystyle \Phi}{2} = \frac{26 \times 18 \times 12/13}{2} = 216$

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*Using Heron's formula for $\triangle BOC$ to find $b$
Defining $OC = a = 26 / 2 = 13$, $OB = b = 18 / 2 = 9$, $BC = c$
$p = \frac{a + b +c}{2} = \frac{13 + 9 + c}{2} = \frac{22 + c}{2}$
$S = \sqrt{p(p-a)(p-b)(p-c)}$
I believe when I find $b$, I will able to find $a$ too, but kinda stuck at that point.
 [/INST] You already noted that the diagonals are divided by half in the center. However, you're making it far too complicated by using the area.
Instead, take a look at the triangle $\triangle AOD$. You already know the length of two sides, $a=9$ and $c=13$ and their inclosed angle $\beta=\operatorname{arcsin} \frac{12}{13}\approx 67.38°$. With the help of the cosine theorem you get:
$$
b^2=a^2+c^2-2ac\cdot\cos \beta \\ 
b = \sqrt{81+169-2\cdot 9 \cdot 13\cdot\cos\left(\operatorname{arcsin} \frac{12}{13}\right)}=4\sqrt{10}\approx 12.65
$$
By using $\angle AOB = 90°-\angle AOD$ you can calculate the length of $a$ analogously within the triangle $\triangle AOB$.
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