2. A right-angled triangle has area $150 \mathrm{~cm}^{2}$ and the length of its perimeter is $60 \mathrm{~cm}$. What are the lengths of its sides?
Solution
Let the sides of the triangle adjacent to the right angle be $a \mathrm{~cm}$ and $b \mathrm{~cm}$. Since the area is $150 \mathrm{~cm}^{2}$ we have $300=a b$.

The length of the hypotenuse, in cm, is $60-(a+b)$, so that, using Pythagoras' theorem, we have
$$
a^{2}+b^{2}=3600-120(a+b)+(a+b)^{2} \text {. }
$$

After expanding and simplifying, we get
$$
120(a+b)=3600+2 a b
$$
and so, substituting $300=a b$, we obtain $a+b=35$. It follows that $a$ and $b$ are roots of the equation $x^{2}-35 x+300=0$, which we may factorise as $(x-20)(x-15)=0$. Therefore $a$ and $b$ are 15 and 20 , in either order.
Hence the sides of the triangle have lengths $15 \mathrm{~cm}, 20 \mathrm{~cm}$ and $25 \mathrm{~cm}$.