The radius of a circle circumscribed about a right-angled triangle is 6.5, and one of its legs is 12. Find the second leg.

We will use the fact that the center of the circle circumscribed about a right-angled triangle lies in the middle of the hypotenuse of this triangle, and the radius of this circle is half the length of the hypotenuse.

In the wording of the condition for this task, it is reported that the radius of the circle described around this right-angled triangle is 6.5, therefore, the length of the hypotenuse of this triangle is 2 * 6.5 = 13.

Since one of the legs of this right-angled triangle is 12, we can find the length of the other leg using the Pythagorean theorem:

√ (13 ^ 2 – 12 ^ 2) = √ ((13 – 12) * (13 +12)) = √ (1 * 25) = √25 = 5.

Answer: the second leg is 5.