4. The cathetus $a$ and the hypotenuse of a right triangle are in the ratio $12: 13$. If we shorten the hypotenuse by $23 \mathrm{~cm}$ and cathetus $a$ by $27 \mathrm{~cm}$, we get a new right triangle whose other cathetus has not changed in length. Calculate the sides of the original triangle.
4. Sides of the first triangle: $a=12 x, b=b, c=13 x$ ..... 0.5 points
Sides of the second triangle: $a^{\prime}=12 x-27$ ..... 0.5 points

$$
\begin{aligned}
b^{\prime} & =b \\
c^{\prime} & =13 x-23
\end{aligned}
$$

0.5 points
Pitagora's theorem written as $b^{2}=c^{2}-a^{2}=c^{\prime 2}-a^{\prime 2}$ ..... 1 point
Data inserted $(13 x)^{2}-(12 x)^{2}=(13 x-23)^{2}-(12 x-27)^{2}$ ..... 1 point
Reformatted and simplified equation $50 x=200$ ..... 1 point
Calculated $x=4$ ..... 0.5 points
Sides of the first triangle: $a=48 \mathrm{~cm}, b=20 \mathrm{~cm}, c=52 \mathrm{~cm}$ ..... 1 point