Triangle abc is reflected across the line y = x. what are the coordinates of the vertex b' of the resulting triangle a'b'c'?

You have not provided the coordinates of the triangle, therefore, I cannot give you a precise answer.
However, I can help you with the concept.

Reflecting a point on the line y = x will lead to the following:
a point having coordinates (x,y) would become a point with coordinates (y,x) when reflected on this line

Examples:
point (2,1) when reflected on y=x would become (1,2)
point (3,7) when reflected on y=x would become (7,3)
point (22,9) when reflected on y = x would become (9,22)

Therefore, for the given point, all you have to is swap the values of the x and y to get the coordinates of the reflected point

Hope this helps :)