What is the orthocenter? triangle ABC has vertices at A (2,3) B (-4,-3) and C(2,3)
a. (-1,-3)
b. (-1,0)
c. (2,-3)
d. (0,0)
The orthocenter of a triangle is the point where the three altitudes of the triangle intersect.
To find the orthocenter of triangle ABC, we need to find the equations of the altitudes first. The slope of a line perpendicular to another line with slope m is -1/m.
The altitude from A will be perpendicular to the line passing through B and C. The slope of the line passing through B(−4,−3) and C(2,3) is (3−(−3))/(2−(−4))=6/6=1. Therefore, the slope of the altitude from A is -1/1=-1. We can find the equation of this altitude using the point-slope form:
y−3=−1(x−2)
Simplifying this equation gives us: y=−x+5
Similarly, the altitude from B will be perpendicular to the line passing through A(2,3) and C(2,3), which is a vertical line x=2.
Finally, the altitude from C will be perpendicular to the line passing through A(2,3) and B(−4,−3). The slope of this line is (3−(−3))/(2−(−4))=6/6=1. Therefore, the slope of the altitude from C is -1/1=-1. We can find the equation of this altitude using the point-slope form:
y−3=−1(x−2)
Simplifying this equation gives us: y=−x+5
To find the orthocenter, we need to solve the system of equations formed by the three altitudes. The system of equations is:
y=−x+5
x=2
y=−x+5
Since x=2 in both equations, the orthocenter of triangle ABC is the point (2, -3).
Thus, the correct answer is c. (2,-3).