Question

I don't understand how to find a. Please help!

Answer 1

The small right angled triangle with hypotenuse vale 5 is similar to the big triangle with b as the hypotenuse
Corresponding sides are in the same ratio so we can write 
a/12 = 5/b.....(1)

Also  applying pythagoras theorem to the large triangle 
b^2 = 5^2 + 12^2
b^2 = 169
b = 13

So substituting in equation (1):-
a  / 12 = 5  / 13
a = (5*12 ) / 13

Answer 2

I wanted to check that answer so I tried a different approach.
From the Pythagorean Theorem, we know that side b = 13
So, (look at my graphic), we can split up side b into "x" and "13-x"
So, we can say
5^2 = a^2 + x^2     12^2 = a^2 + (13 -x)^2
25 - x^2 = a^2        a^2 = -25 + 26x -x^2
Therefore,
25 - x^2 = -25 + 26x -x^2
26x = 50
 x = 1.9230769231
If x = 1.9230769231 then
a^2 = 5^2 -1.9230769231^2
a^2 = 25 -3.6982248521
a^2 = 21.3017751479
a = 4.6153846154