In right triangle ΔUVW with ∠V = 90 degrees, if cosec U = 13/12 and side UV = 2.5 cm, use trigonometric ratios and Pythagoras theorem to find the length of side VW (in centimeters) and choose the correct option.

Introduction / Context:
This question links trigonometric ratios with side lengths in a right triangle. You are given the value of cosec U and the length of one side UV, and you must determine the length of another side VW. The angle at V is the right angle, so Pythagoras theorem can be used along with the definition of cosec U to find the missing side.

Given Data / Assumptions:

• Right triangle ΔUVW has ∠V = 90 degrees.
• cosec U = 13/12.
• Side UV = 2.5 cm.
• All side lengths are positive real numbers.

Concept / Approach:
For angle U, sine is defined as opposite side divided by hypotenuse, so cosec U = hypotenuse / opposite side. In this triangle, the hypotenuse is UW because the right angle is at V. The side opposite angle U is VW. From cosec U = 13/12 we can write the ratio UW/VW = 13/12, which allows us to express UW and VW in terms of a common scale factor. Then Pythagoras theorem relates UV, VW, and UW, and we use the given UV = 2.5 cm to solve for the scale factor and thus for VW.

Step-by-Step Solution:
From cosec U = 13/12, write UW/VW = 13/12. Let UW = 13k and VW = 12k for some positive constant k. Since ∠V = 90 degrees, use Pythagoras theorem: UV^2 + VW^2 = UW^2. Substitute UV = 2.5, VW = 12k, and UW = 13k to get (2.5)^2 + (12k)^2 = (13k)^2. Compute squares: 6.25 + 144k^2 = 169k^2, so 6.25 = 25k^2. Thus k^2 = 6.25 / 25 = 0.25 and k = 0.5 (taking the positive root). Now find VW = 12k = 12 * 0.5 = 6 cm.
Verification / Alternative check:
You can also compute UW = 13k = 13 * 0.5 = 6.5 cm and confirm that UV^2 + VW^2 = UW^2 becomes 2.5^2 + 6^2 = 6.5^2, that is 6.25 + 36 = 42.25 and 6.5^2 = 42.25, which matches exactly. This confirms that the triangle with sides 2.5 cm, 6 cm, and 6.5 cm is consistent with the given cosec value.

Why Other Options Are Wrong:
Option a ( 4 cm ), option c ( 5.6 cm ), option d ( 6.5 cm ), and option e ( 7.5 cm ) do not satisfy both the Pythagoras relation and the ratio UW/VW = 13/12 when UV = 2.5 cm. They lead to inconsistent triangle side lengths or incorrect cosec values for angle U.

Common Pitfalls:
One common mistake is to treat UV as the hypotenuse instead of UW, which breaks the interpretation of cosec U. Another error is to apply Pythagoras theorem with the wrong combination of sides or to forget to square the numeric values correctly. Keeping the geometry of the right triangle clear avoids these issues.

Final Answer:
The length of side VW in the right triangle is 6 cm.