Which one of the following set of internal angles (degree) of a triangle does show a well-conditioned triangle?
Correct Answer:
(D) 35, 80, 65
📝 Detailed Explanation: Well-Conditioned vs. Ill-Conditioned Triangles
In surveying, especially in triangulation, the shape of the triangles is crucial for maintaining accuracy. For precise plotting and measurement, it is desirable to use "well-conditioned" or "well-shaped" triangles, as this minimizes the propagation of errors.
What is a Well-Conditioned Triangle?
The Ideal Shape
A triangle is considered well-conditioned if all its internal angles are within a specific range.
- No angle should be less than 30°.
- No angle should be greater than 120°.
- The best possible shape is an equilateral triangle (all angles 60°), as it is perfectly symmetrical.
Ill-Conditioned Triangles: The Problem
A triangle with an angle smaller than 30° or larger than 120° is "ill-conditioned."
- Very small angles (< 30°): The intersection of the two sides is poorly defined. A small error in measuring either side can cause a large error in the location of the vertex.
- Very large angles (> 120°): This automatically creates two other small angles in the triangle, leading to the same problem of poor intersection.
⚙️ Analyzing the Options
Based on the rule (angles must be between 30° and 120°), we can analyze each option:
- (a) 20, 90, 70: This is an ill-conditioned triangle because the 20° angle is less than 30°.
- (b) 25, 45, 110: This is an ill-conditioned triangle because the 25° angle is less than 30°.
- (c) 40, 125, 15: This is an ill-conditioned triangle for two reasons: the 125° angle is greater than 120°, and the 15° angle is less than 30°.
- (d) 35, 80, 65: This is a well-conditioned triangle. All angles are safely within the 30° to 120° range.
Conclusion: The question asks which set does show a well-conditioned triangle. Only option (d) has all its angles between 30° and 120°. Therefore, it is the only well-conditioned triangle among the choices.