The orthocenter of △ABC is at the point
. Option D is the correct choice.
The orthocenter of a triangle is the point where the altitudes intersect. To find the coordinates of the orthocenter of triangle ABC, we can first find the equations of the altitudes.
The altitude from A to BC has slope and passes through A (2, 4), so its equation is y−4=−1(x−2). Solving for y, we get y=−x+6.
The altitude from B to AC has slope
and passes through B (4, 2), so its equation is y−2=1(x−4).
Solving for y, we get y = x−2
Setting these two equations equal to each other, we get −x+6=x−2, which means x=4. Substituting this value into either equation, we get y=2. Therefore, the altitudes intersect at the point (4, 2).
However, this is not the orthocenter of triangle ABC. The orthocenter is actually the foot of the altitude from C to AB. To find the equation of this altitude, we first need to find the slope of AB. AB has slope
, so the altitude from C to AB has slope 1 (perpendicular slopes). We also know that the altitude passes through C (-2, -4).
Using the point-slope form of linear equations, we can find that the equation of the altitude from C to AB is y+4=1(x+2). Solving for y, we get y=x−2.
Setting this equation equal to the equation of the line containing A and B (y = -x + 6), we get x−2=−x+6. Solving for x, we get x=7. Substituting this value into either equation, we get y=−5. Therefore, the orthocenter of triangle ABC is (7, -5).
So the answer is D. (7, -5).