Question

Two congruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is 8: 7. Find the minimum possible value of their common perimeter.

Answer 1

Let S be the semi-perimeter of the two triangles. Also, let the base of the longer triangle be 16x and the base of the shorter triangle be $14x$ for some arbitrary factor x Then, the dimensions of the two triangles must be s-8x,s-8x,16x and s-7x,s-7x,15x. By Heron's Formulae, we have

$\sqrt{s(8x)(8x)(s-16x)}=\sqrt{s(7x)(7x)(s-14x)}$

$8\sqrt{s-16x}=7\sqrt{s-14x}$

$64s-1024x=49s-686x$

$15s=338x$

Since $15$ and $338$ are coprime, to minimize, we must have $s=338$ and $x=15$. However, we want the minimum perimeter. This means that we must multiply our minimum semiperimeter by $2$, which gives us a final answer of $676$.