What is the length of the altitude drawn to the hypotenuse

You can make 3 equations based on the multiple right triangles, the pythagorean theorem and the knowledge that 7 + 15 is 22. (I am labeling the unknown left side of the triangle as x and the unknown right side as y) 
--> the altitude has been labeled a
1. (22)squared = (x)squared + (y)squared
2. (x)squared = (7)squared + (a)squared
3. (y)squared = (15)squared + (a)squared

To find the necessary components for 22 squared (x2 + y2), you can add the 2 formulas for xsquared and ysquared together to get:
(x)squared + (y)squared = (7)squared + (15) squared + 2(a)squared
This can be simplified to : (x)squared + (y) squared = 274 + 2(a)squared

Since 274+2(a)squared equals x2+y2, substitute this answer in for your first equation : 22 squared = 274 + 2(a)squared
simplify -- > 484 = 274 + 2(a)squared --> 210 = 2(a)squared 
 --> 105 = (a)squared --> square root of (105) = a