In triangle abc, a is a right angle, and mb = 45°. find bc. if your answer is not an integer, leave it in simplest radical form.

the diagram is not drawn to scale.

-- In a right triangle, if ∠B is 45°, then ∠C is too.

-- Since ∠B  and  ∠C  are equal, the sides opposite them
are equal too.  This is an isosceles right triangle.  Its legs
are equal.

-- (The hypotenuse)²  =  (one leg)²  +  (the other leg)²

                                   =       (12-ft)²  +  (12-ft)²

                                   =              (144 x 2)

      The hypotenuse  =  √(144 x 2)  =  12√2 .

Since A is a right angle and mB = 45°, the right triangle is a 45-45-90 triangle.

This means the following:
long leg and short leg have the same value of x
hypotenuse is equal to x√2

Since the long leg has been given as 12 ft. The measure of the short leg is also 12 ft. 

hypotenuse = x√2 = 12 √2 = 12 * 1.414 = 16.97 ft  ⇒ this is the measure of BC.

One that half its volume