In right-angled triangle UVW, with angle V = 90°, if cosec U = 13/12 and side UV = 2.5 cm, what is the length (in cm) of side VW opposite angle U?

Introduction / Context:
This question combines trigonometric ratios with right triangle geometry and a real length. You are given cosec U and one side length, and you must find another side of the right triangle. It tests understanding of how cosec relates to the ratio of hypotenuse to opposite side, and also how to recognise standard Pythagorean triples.

Given Data / Assumptions:

• Triangle UVW is right-angled at V, so angle V = 90°.
• cosec U = 13/12.
• Side UV has length 2.5 cm.
• We must find the length of side VW, which is opposite angle U.

Concept / Approach:
By definition, cosec U = hypotenuse / opposite. In triangle UVW, the side opposite angle U is VW, and the hypotenuse is UW. If cosec U = 13/12, then we can take UW = 13k and VW = 12k for some positive scale factor k. The remaining side UV is adjacent to angle U and must satisfy the Pythagorean triple relation, giving UV = 5k in a 5, 12, 13 triangle. Since UV is given as 2.5 cm, we can find k and then compute VW = 12k.

Step-by-Step Solution:
From cosec U = 13/12, set hypotenuse UW = 13k and opposite side VW = 12k.In a 5, 12, 13 right triangle, the third side is 5k. Here that side corresponds to UV.Thus UV = 5k. We are given UV = 2.5 cm.So 5k = 2.5, which gives k = 2.5 / 5 = 0.5.Now compute VW = 12k = 12 * 0.5 = 6 cm.Therefore, the length of side VW is 6 cm.

Verification / Alternative check:
Check consistency with cosec U. With k = 0.5, UW = 13k = 6.5 cm and VW = 6 cm. Then sin U = opposite / hypotenuse = VW / UW = 6 / 6.5 = 12 / 13. Hence cosec U = 1 / sin U = 13 / 12, which matches the given value. Also, confirm the Pythagorean theorem: UV^2 + VW^2 = 2.5^2 + 6^2 = 6.25 + 36 = 42.25, and UW^2 = 6.5^2 = 42.25. This verifies that the sides form a valid right triangle.

Why Other Options Are Wrong:

• 6.5 is the hypotenuse, not the opposite side VW.
• 4, 5, and 5.6 do not maintain the correct cosec U value or the 5, 12, 13 ratio when combined with UV = 2.5.
• Only 6 cm produces both the correct triangle shape and the given cosec U.

Common Pitfalls:

• Misidentifying which side is opposite angle U and which is the hypotenuse in the right triangle.
• Forgetting that once you introduce k in the triple 5, 12, 13, all sides must be scaled by the same factor.
• Using the wrong trigonometric ratio (for example, treating cosec as if it were sec or something else).

Final Answer:
6