Problem Statement
A large pond obstructs the chain line AB. To circumvent the obstacle, a line AL was measured on the left of AB (length 901 m), and another line AM was measured on the right of AB (length 1100 m). The points L, B, and M lie on the same straight line. The measured length of BL is 502 m, and the length of BM is 548 m. The task is to find the obstructed distance AB.
Step-by-Step Solution
Key Information
- Chain line AB is obstructed by a pond.
- Auxiliary line AL = 901 m.
- Auxiliary line AM = 1100 m.
- Points L, B, M are collinear (on the same straight line).
- Segment BL = 502 m.
- Segment BM = 548 m.
- Goal: Find the distance AB.
Step 1: Calculate the Total Length LM
Since points L, B, and M are on the same straight line, the total length LM is the sum of lengths BL and BM.
LM = BL + BM
LM = 502 m + 548 m
LM = 1050 m
Step 2: Apply Law of Cosines to Triangle ALM
Consider the triangle ALM. We know the lengths of all three sides: AL = 901 m, AM = 1100 m, and LM = 1050 m. We can use the Law of Cosines to find the cosine of the angle at M (let’s call it θ = ∠AML).
cos(θ) = (AM2 + LM2 – AL2) / (2 * AM * LM)
cos(θ) = (11002 + 10502 – 9012) / (2 * 1100 * 1050)
cos(θ) = (1,210,000 + 1,102,500 – 811,801) / (2,310,000)
cos(θ) = 1,500,699 / 2,310,000
cos(θ) ≈ 0.649653
Step 3: Apply Law of Cosines to Triangle AMB
Now consider triangle AMB. We know AM = 1100 m and BM = 548 m. The angle at M (∠AMB) is the same angle θ calculated in Step 2 because L, B, and M are collinear. We want to find the length of side AB. Apply the Law of Cosines again:
cos(θ) = (AM2 + BM2 – AB2) / (2 * AM * BM)
cos(θ) = (11002 + 5482 – AB2) / (2 * 1100 * 548)
cos(θ) = (1,210,000 + 300,304 – AB2) / (1,205,600)
cos(θ) = (1,510,304 – AB2) / 1,205,600
Step 4: Equate Expressions for cos(θ) and Solve for AB
We have two expressions for cos(θ) from Step 2 and Step 3. Set them equal to each other:
(1,500,699 / 2,310,000) = (1,510,304 – AB2) / 1,205,600
Now, solve for AB2:
1,510,304 – AB2 = (1,500,699 / 2,310,000) * 1,205,600
1,510,304 – AB2 ≈ 0.649653 * 1,205,600
1,510,304 – AB2 ≈ 783,221.95
AB2 ≈ 1,510,304 – 783,221.95
AB2 ≈ 727,082.05
Finally, take the square root to find AB:
AB = √727,082.05
AB ≈ 852.69 m
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Chain Line Obstruction: Situations in surveying where a direct measurement along a line is blocked (e.g., by a pond, building).
- Circumventing Obstacles: Using auxiliary lines and geometric principles to calculate the obstructed distance indirectly.
- Collinear Points: Points that lie on the same straight line (L, B, M in this case).
- Law of Cosines: A fundamental trigonometric rule relating the lengths of the sides of a triangle to the cosine of one of its angles (c² = a² + b² – 2ab cos(C)). Crucial for solving triangles when angles aren’t right angles.
Real-World Applications:
- Surveying across rivers, lakes, ponds, or heavily vegetated areas.
- Calculating distances between points when direct access is impossible due to buildings or terrain.
- Used in land surveying, civil engineering, and mapping to establish property boundaries or plan construction routes.
Why It Works:
This method works by creating two triangles (ΔALM and ΔAMB) that share a common angle (∠AMB = ∠AML = θ). By measuring the sides of the larger triangle ALM (where LM is found by adding BL and BM since L, B, M are collinear), we can calculate the cosine of the shared angle θ using the Law of Cosines. Then, focusing on the smaller triangle AMB, we again apply the Law of Cosines. This time, we know two sides (AM and BM) and the cosine of the included angle (cos θ from the first calculation). This allows us to set up an equation where the only unknown is the length of the side opposite the angle, which is the required distance AB. Equating the two derived expressions for cos(θ) provides the path to solve for AB.