2025 SJI P2 Q7

On a hillside, a surveyor measured a triangular plot PQR with

\[PQ=10\text{ m},\quad QR=x\text{ m},\quad PR=y\text{ m},\quad\text{and}\hspace{0.5em}\angle PQR=30^\circ.\]

1. Show that \(x^2-10\sqrt3x+100-y^2=0\).[3]
2. Consider the possible triangles with \(y=8\).
   1. Calculate the two possible values of \(x\).
   2. Hence find the area of the smaller of the two triangles.[4]
3. Consider the case where the length of PR, \(y\) m, is not fixed at 8 m. Find the range of values of \(y\) for which it is possible to form two triangles.[6]