A (3,-1), B (0,0) and C (-3,k) are the vertices of a right-angled triangle with right angle at B. Find the value of k.

To find the value of k, we need to determine the slope of line segment BC and find where it intersects the y-axis.

The slope of line segment BC is given by (change in y ÷ change in x) = (k - 0) ÷ (-3 - 0) = k/(-3).
Since line segment BC is perpendicular to line segment AB (the hypotenuse), the product of their slopes is -1.
So, the slope of line segment AB is the negative reciprocal of the slope of line segment BC.
The slope of line segment AB is given by (-1 - 0) ÷ (3 - 0) = -1/3.
Hence, the slope of line segment BC is the negative reciprocal of -1/3, which is 3.
Now we have: k/(-3) × 3 = -1.
=> k/(-1) = -1.
=> k = 1.
Therefore, the value of k is 1.