Problem

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Figure 1 shows a sketch of a triangle $LMN$ and a sketch of a triangle $PQR$.

Figure 1

Triangle $LMN$ is an equilateral triangle with sides $(x+7)$ cm.

Triangle $PQR$ is a right-angled isosceles triangle with $PQ=PR=2x$ cm and angle $QPR=90^\circ$.

(a) Find, in its simplest form in terms of $x$, the length $QR$.

Given that the perimeter of triangle $LMN$ is equal to the perimeter of triangle $PQR$,

(b) find the value of $x$, writing your answer in the form $a\sqrt2+b$, where $a$ and $b$ are integers.