In a three-hinged arch analysis by funicular/graphical construction, measured distances are AD = 4 cm, DD' = 3 cm, and AD' = 5 cm. Determine the angle between the reaction directions at supports A and B (choose the nearest correct option).

Introduction / Context:
Graphical methods for arches and frames use force polygons and funicular constructions. Distances measured on the load line and parallel lines relate to the magnitudes and directions of reactions. This problem uses a simple geometric relation emerging from a right-triangle measurement.

Given Data / Assumptions:

• Measured lengths: AD = 4 cm, DD' = 3 cm, AD' = 5 cm.
• Standard three-hinged arch force polygon construction is assumed.
• The angle between reaction directions at A and B is sought.

Concept / Approach:
The measured segments form a triangle with sides 3, 4, and 5. Recognition of the 3–4–5 Pythagorean triplet indicates a right angle in the associated force polygon. In this classical construction, the right angle corresponds to the angle between the two reaction lines, yielding an orthogonal (90°) relationship.

Step-by-Step Solution:
1) Identify triangle with sides AD = 4, DD' = 3, and AD' = 5.2) Check Pythagoras: 4^2 + 3^2 = 16 + 9 = 25 = 5^2 → right triangle.3) Map this right angle in the force polygon to the included angle between reaction vectors.4) Conclude the reaction directions are mutually perpendicular.

Verification / Alternative check:
Alternate geometric derivations (cosine law on the force triangle) also yield cos(theta) = 0 when the measured lengths satisfy a 3–4–5 relation.

Why Other Options Are Wrong:

• 30°, 45°, 60°: Not supported by the 3–4–5 right-triangle result; these would require different length ratios.

Common Pitfalls:

• Misidentifying which included angle corresponds to the reaction directions in the force polygon.
• Arithmetic slips when checking Pythagoras.

Final Answer:
90°.