Since both A and B are the same measure of angle, then you have an isosceles triangle so sides a and b have the same length. But there's another advantage, because the third angle is a right angle, so you can use Pythagoras' Theorem: c2=a2+b2, and since you know a=b, this becomes 2a2=c2. Take the square root of each side and we get a√2=c, so a=c/√2. You are given c=10.16, therefore a (and b)=10.16/√2=7.18 approx. This solution doesn't depend on trigonometry but it only works with right triangles.
You can solve using trigonometry. a/c=sinA=sin(45)=1/√2. Therefore a (and b)=c/√2=7.18 as before.
You can also use the Sine Rule because you have two angles and a side:
a/sinA=b/sinB=c/sinC; a/sin(45)=b/sin(45)=c/sin(90)=10.16/1=10.16. And we arrive at the same answer again, because we know that sin(45)=1/√2.
You can always use the Sine Rule if you have one side and two angles, and you need to find the length of another side.
So there are a few formulas you can use to solve this problem.
The Cosine Rule is another way if you have two sides and the included angle (the angle between the known sides).
To summarise: Pythagoras for right triangles; Sine Rule or Cosine Rule when you don't have a right triangle.