Find the value of k in the following figure.

From the secant theorem we know that:

3(k+3)=5(5+6)

3k+9=55

3k=46

k = = 15 1/3

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.