To solve this problem, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Let's call the shorter side of the original triangle "a" and the longer side "b". According to the problem, the hypotenuse (c) is 10 cm. We can set up the equation as follows:

a^2 + b^2 = c^2
a^2 + b^2 = 10^2
a^2 + b^2 = 100 ----(1)

The area of the original triangle is given as 24 cm^2. The formula for the area of a right-angled triangle is (1/2) * base * height. Since one of the sides is the base and the other is the height, we can write:

(1/2) * a * b = 24
a * b = 48 ----(2)

Now, let's consider the new triangle. According to the problem, the shorter side (a) is halved and the longer side (b) is doubled. So the new sides are (1/2) * a and 2 * b. Let's call the new hypotenuse "d".

Using the same logic, we can set up the equation for the new triangle as follows:

(1/2 * a)^2 + (2 * b)^2 = d^2
(1/4) * a^2 + 4 * b^2 = d^2
(1/4) * a^2 + 4 * b^2 = d^2 ----(3)

Now, we need to find the value of d. From equation (1), we can rewrite it as:

a^2 = 100 - b^2

Substituting this into equation (3), we get:

(1/4) * (100 - b^2) + 4 * b^2 = d^2
25 - (1/4) * b^2 + 4 * b^2 = d^2
(15/4) * b^2 + 25 = d^2 ----(4)

Now, we can solve equations (2) and (4) simultaneously to find the value of b and d. We'll substitute the value of a from equation (2) into equation (4):

(15/4) * (48/b) + 25 = d^2
(720/b) + 25 = d^2
(720 + 25b)/b = d^2 ----(5)

We can also substitute the value of a from equation (2) into equation (1):

(48/b)^2 + b^2 = 100
2304/b^2 + b^2 = 100
2304 + b^4 = 100b^2
b^4 - 100b^2 + 2304 = 0 ----(6)

Now, we have two equations (5) and (6) that we can solve simultaneously to find the values of b and d.

By solving these equations, we find that b = 4 cm and d = 8 cm.

Therefore, the new hypotenuse becomes 8 cm.