---
title: "Length of PQ across a pond"
subject: "Surveying I"
chapter: 3
key: "semester_3/surveying_i:obstacle-pond-chain-line"
source: https://www.bctengineer.com/q/semester_3/surveying_i/obstacle-pond-chain-line
site: BCT Engineering
---

# Length of PQ across a pond

Surveying I · Chapter 3: Chain Survey

- Asked in: 2062 Poush, 8 marks
- Video: [Chain surveying: finding width of river-1](https://www.youtube.com/watch?v=hXarAXOojKc) — civil engineering basics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/obstacle-pond-chain-line
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-3

## Question

A chain line AB intersects a pond. Two points P and Q are taken on the chain line on opposite sides of the pond. A line PR, 230 m long, is set out on the left of PQ and another line PS, 280 m long, is set out on the right of PQ. Points R, Q and S are in the same straight line. RQ and QS are 80 m and 130 m long respectively. Calculate the length of PQ.

## Short answer

Applying Stewart's theorem to triangle PRS with Q on RS: PR²·QS + PS²·RQ = RS(PQ² + RQ·QS). Substituting values gives PQ² = 52,214.29, so PQ = 228.50 m.

## Answer

#### 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.

```text
            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.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Which theorem finds PQ from the cevian in triangle PRS?
   A. Pythagoras only
   B. Simpson's rule
   C. Stewart's theorem
   D. Bowditch's rule
   Answer: C. Q lies on side RS and all other lengths are known. (Key point: Theorem used)

2. Length RS equals:
   A. 50 m
   B. 80 m
   C. 130 m
   D. 210 m
   Answer: D. RS = RQ + QS = 80 + 130. (Key point: Length RS)

3. Stewart's equation here is:
   A. PR^2 x QS + PS^2 x RQ = RS (PQ^2 + RQ x QS)
   B. PR^2 x RQ + PS^2 x QS = RS (PQ^2 + RQ x QS)
   C. PR^2 + PS^2 = PQ^2
   D. PQ = PR - PS
   Answer: A. Each outer side squared is multiplied by the segment opposite it. (Key point: Equation)

4. PQ is approximately:
   A. 250 m
   B. 228.5 m
   C. 210 m
   D. 52,214 m
   Answer: B. PQ^2 = 52,214.29, so PQ is about 228.5 m. (Key point: Result)

## Related in Surveying I

- [Chainage of R across a river](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aobstacle-river-chainage-of-r) — Surveying I, chapter 3
- [True length of base line AB (four bays)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-base-line-four-bays) — Surveying I, chapter 2
- [True length: tape found 2.5 cm too long](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-825-plus-750-m) — Surveying I, chapter 2
- [Area enclosed by chain line and hedge](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-hedge-5m-interval-simpson-trapezoidal) — Surveying I, chapter 9
- [Two peg test, AB = 40 m, near B](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atwo-peg-ab-40m-inner-side-b) — Surveying I, chapter 5
- [True length of suspended distance AB (66)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Atape-suspended-ab-114-095) — Surveying I, chapter 2

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