The radius of the circle hanging in a right-angled triangle is 4 and one of the legs is 12 find the length of the other leg.

Let x denote the length of the second leg of this triangle.

Since the length of the first leg is 12, then, according to the Pythagorean theorem, the hypotenuse of this triangle should be equal to √ (12 ^ 2 + x ^ 2) = √ (144 + x ^ 2), and the area of ​​this triangle should be equal to 12x / 2 = 6x.

According to the condition of the problem, the radius of a circle inscribed in this right-angled triangle is 4, therefore, applying the formula for the area of ​​a triangle through the radius of the inscribed circle, we can compose the following equation:

6x = 4 * (12 + x + √ (144 + x ^ 2)) / 2

solving which, we get:

6x = 2 * (12 + x + √ (144 + x ^ 2));

12 + x + √ (144 + x ^ 2) = 6x / 2;

12 + x + √ (144 + x ^ 2) = 3x;

√ (144 + x ^ 2) = 3x – x – 12;

√ (144 + x ^ 2) = 2x – 12;

144 + x ^ 2 = (2x – 12) ^ 2;

144 + x ^ 2 = 4x ^ 2 – 48x + 144;

x ^ 2 = 4x ^ 2 – 48x;

4x ^ 2 – 48x – x ^ 2 = 0;

3x ^ 2 – 48x = 0;

3x * (x – 16) = 0;

x1 = 0;

x2 = 16.

Since the leg length cannot be equal to 0, the value x = 0 is not suitable.

Therefore, the length of the other leg is 16.

Answer: 16.