What are two different ways you could find the value of a? explain these methods.

Using the Pythagorean theorem we can know from the first triangle that the height of the two triangles are 12. So, we then know that a = 20. 
(Root 15^2-9^2=12) (Root 12^2+16^2=20)

Another way is using the whole triangle. The step above used two triangles. The two triangles together makes a big right triangle, so we can again use the Pythagorean theorem. So, the answer again is 20. 
(Root 25^2-15^2=20)

You could use the Pythagorean theorem, since you know the length of the hypotenuse is 9 + 16 = 25 units and the length of one leg is 15 units. To find the value of a, use the relationship that the the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. You could also use the geometric mean (leg) theorem, which states that the length of the hypotenuse is to the length of an adjacent leg as that adjacent leg length is to the length of its corresponding segment in the hypotenuse. So you could write and solve the proportion 25/a = a/6.