Length of PQ across a pond

Data

Points R, Q and S are collinear with RQ = 80 m, QS = 130 m, so RS = 210 m. P is on the chain line, with PR = 230 m and PS = 280 m. PQ is required.

            P
           /|\ 
      230 / | \ 280
         /  |  \ 
        R---Q---S
         80  130

Stewart's theorem

For point Q on side RS of triangle PRS:

$$PR^{2}\times QS + PS^{2}\times RQ = RS\,(PQ^{2} + RQ\times QS)$$ $$230^{2}\times 130 + 280^{2}\times 80 = 210\,(PQ^{2} + 80\times 130)$$ $$52900\times 130 + 78400\times 80 = 6{,}877{,}000 + 6{,}272{,}000 = 13{,}149{,}000$$ $$PQ^{2} + 10{,}400 = \frac{13{,}149{,}000}{210} = 62{,}614.29$$ $$PQ^{2} = 52{,}214.29,\qquad PQ = 228.50\ \text{m}$$

Check using the cosine rule

$\cos R = \dfrac{230^{2} + 210^{2} - 280^{2}}{2\times 230\times 210} = 0.19255$, so

$$PQ^{2} = 230^{2} + 80^{2} - 2\times 230\times 80\times 0.19255 = 52{,}214.29$$

Answer: PQ = 228.50 m.