The legs of a right-angled triangle are 9 and 12. Find the ratio of the median to the height drawn to the hypotenuse.

1. Let the legs are known in a right-angled triangle ABC with hypotenuse AB:

AC = 9;
BC = 12.
2. Draw the height CH and the median CM to the hypotenuse. The median CM is half the hypotenuse:

AB ^ 2 = AC ^ 2 + BC ^ 2 = 9 ^ 2 + 12 ^ 2 = 81 + 144 = 225;
AB = 15;
MC = 15/2 = 7.5.
3. The area of a right-angled triangle ABC is:

S = 1/2 * AC * BC;
S = 1/2 * AB * CH, hence:
AC * BC = AB * CH;
CH = AC * BC / AB = 9 * 12/15 = 36/5 = 7.2.
4. Ratio of median and height:

CM / CH = 7.5 / 7.2 = 75/72 = 25/24.

Answer: 25/24.