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**HallsofIvy** The lines x+ y+ 1= 0, and y= k intersect where x+ k+ 1= 0 or x= -k- 1. That is the point (-k- 1, k).

The lines y= 2x- 1 and y= k intersect where 2x- 1= k or x= (k+1)/2. That is the point ((k+1)/2, k).

The lines x+ y+ 1= 0 and y= 2x- 1 intersect where x+ 2x-1+ 1= 3x= 0. That is the point (0, -1)

Choose the horizontal line y= k as base. The distance from (1- k, k) to (k+1, k) is k+ 1- (1- k)= 2k. That is the length of the base.

The altitude is the distance from the third vertex, (0, -1), to the base, measured along the line perpendicular to the base. Since the base is y= k, the perpendicular is x= 0. That crosses y= k at (0, k) which has distance from (0, -1) k- (-1)= k+ 2 and that is the "height" of the triangle. I presume you know that the area of a trangle is "1/2 base times height".