First of all, let's find the dimensions of your triangle. The middle length of the ratio is 4, so the scale factor is (12 cm)/4 = 3 cm. Thus the other two side lengths are 3×3 cm = 9 cm, and 5×3 cm = 15 cm.
The area of a triangle is the same no matter how you compute it. If you compute it using the legs of the right triangle, you find it is
A = (1/2)×(12 cm)×(9 cm) = 54 cm²
If you compute it using the hypotenuse and the height to the hypotenuse, it is the same.
A = 54 cm² = (1/2)×(15 cm)×height
We solve this for height by dividing by the coefficient of height. Doing so, we find
height = (54 cm²)/(7.5 cm) = 7.2 cm
The height to the hypotenuse is 7.2 cm.
