In a right angled triangle, the hypotenuse is 10 cm and the area is 24 square centimetres. If the shorter leg is halved and the longer leg is doubled, what is the length of the new hypotenuse in centimetres?

Introduction / Context:
This problem requires finding the original legs of a right angled triangle from its hypotenuse and area, and then modifying those legs to form a new triangle. After the shorter leg is halved and the longer leg is doubled, we must compute the new hypotenuse. The question tests algebraic reasoning with products and sums, right triangle relationships, and careful manipulation of the side lengths.

Given Data / Assumptions:

• Original hypotenuse c = 10 cm.
• Area of original triangle = 24 sq cm.
• For a right triangle with legs a and b: c^2 = a^2 + b^2.
• Area = (1/2) * a * b.
• Shorter leg is halved, longer leg is doubled to form new triangle.

Concept / Approach:
From the area, we get a*b = 48, because (1/2)*a*b = 24. From the hypotenuse, we know a^2 + b^2 = 100. Using the identities for sum and product of roots, we can determine a and b. Then we identify the shorter and longer legs, transform them as instructed, and finally apply the Pythagoras theorem again to compute the new hypotenuse.

Step-by-Step Solution:
From area: (1/2)*a*b = 24, so a*b = 48.From hypotenuse: a^2 + b^2 = 10^2 = 100.Compute (a - b)^2 = a^2 + b^2 - 2ab = 100 - 96 = 4.Therefore |a - b| = 2 and a + b = sqrt(a^2 + b^2 + 2ab) = sqrt(100 + 96) = sqrt(196) = 14.Solving a + b = 14 and a - b = 2 gives a = 8, b = 6 (or vice versa).Shorter leg = 6 cm, longer leg = 8 cm.New shorter leg = 6 / 2 = 3 cm.New longer leg = 8 * 2 = 16 cm.New hypotenuse^2 = 3^2 + 16^2 = 9 + 256 = 265.New hypotenuse = sqrt(265) cm.

Verification / Alternative check:
The original sides 6, 8, and 10 form a well known Pythagorean triple, so the initial data is consistent. After transformation, we check that 3 and 16 still form the legs of a right triangle. The computed hypotenuse squared is 265, which does not factor into a perfect square, so leaving the answer as sqrt(265) is appropriate. This matches the expected format in the options.

Why Other Options Are Wrong:
Values such as sqrt(245), sqrt(255), and sqrt(275) arise from using incorrect transformations of the legs or arithmetic mistakes in squaring and adding. sqrt(225) corresponds to 15, which would require 9 and 12 as legs but does not follow the given transformation rules. Only sqrt(265) is consistent with halving 6 to 3 and doubling 8 to 16.

Common Pitfalls:
Learners may mix up which leg is shorter, apply both operations to the same leg, or forget that area (1/2)*a*b is given, not a*b directly. Another typical error is to treat a and b as 7 and 7, which would not satisfy ab = 48. Careful algebra with products and sums avoids these mistakes and leads to the correct new hypotenuse.

Final Answer:
√265 cm