(a) The sum of two numbers is 34. If 3 is subtracted from one number and 2 is added to another, the product of these two numbers becomes 260. Find…

Let the shortest side be \(\displaystyle x \mathrm{~cm}\) \[\left.\therefore \begin{array}{l} \text { Hypotenuse is } 2 x+6 \mathrm{~cm} \\ \text { and other side is } 3 x-6 \mathrm{~cm} \end{array}\right\} \] Hence \[\left.\begin{array}{l} \quad(2 x+6)^{2}=x^{2}+(3 x-6)^{2} \\ \Rightarrow 6 x^{2}-60 x=0 \end{array}\right\} \] \[\Rightarrow x=10 \] Dimensions of the triangle are $\displaystyle 10$ cm, $\displaystyle 24$ cm and $\displaystyle 26$ cm.