Find the orthocenter for the triangle described by each set of vertices. K (4, −4), L (2, 2), M (5, −4) The coordinates of the orthocenter are?

Answer 1

The slope from (2,2) to (5,-4) is m=-2. The slope of the altitude to that side is the negative reciprocal which is 1/2. We can find the equation of the line through that altitude can be found using the slope of the altitude and the point (2,-5). Substituting the coordinates (4,-4) and the slope of 1/2 into y=mx+b produces -4=(1/2)4 + b. Solve for b to get -6. The equation of the line through the altitude is y = (1/2)x -6. The altitude from the point (2,2) to the line through the points (4,-4) and (5,-4) has an undefined slope because it is a vertical line. Therefore its equation is x=2. Substituting x=2 into the first equation produces y= (1/2)2 - 6 = -5. Therefore the intersection of the two altitudes occurs at the point (2,-5)

Answer 2

I don't know how to draw pictures or use latex here but the answer I got was: x=3.667, y=-1.02. Let me know if this is correct (I haven't checked) and maybe I can find some way to walk through this.