14. Each side of an isosceles triangle is a whole number of centimetres. Its perimeter has length $20 \mathrm{~cm}$. How many possibilities are there for the lengths of its sides?
A 3
B 4
C 5
D 6
E 7
Solution: B
Let the length of the two equal sides of the isosceles triangle be $a \mathrm{~cm}$. Since the triangle has perimeter $20 \mathrm{~cm}$, the third side will have length $20-2 a \mathrm{~cm}$. Since this must be a positive length, $20-2 a>0$ and so $a<10$. In a triangle the length of one side must be less than the sum of the lengths of the other two sides. So $20-2 a<2 a$. This gives $20<4 a$ and hence $5<a$. So, we have $5<a<10$. Therefore, as $a$ is a whole number, there are just four possible values for $a$, namely 6 , 7,8 and 9 . So there are four possibilities for the side lengths of the triangle,
$$
6,6,8 ; \quad 7,7,6 ; \quad 8,8,4 \text { and } 9,9,2 \text {. }
$$