Eight isosceles right-angled triangles, whose sides coincide as shown, form a shape whose perimeter is an irregular nonagon. The shortest side of the nonagon has length 1cm . Calculate the perimeter of the nonagon, giving your answer in the form (a)b +(square root))cm .
![](https://amsp.org.uk/app/uploads/2022/08/isoceles-1.png)
The easiest approach to this question is to label each side with a length. It is important to notice that, because every triangle is similar, the ratios of their sides are the same for every triangle – so the hypotenuse is always (square root equation) times the length of the shorter sides.
![](https://amsp.org.uk/app/uploads/2022/08/isoceles-2.png)
Now all that’s left to do is add all the lengths up, adding the integers and surds separately.
1 + 2 + 4 + 8 + 15 = 30
(square root equation)
(square root equation)