9514 1404 393
Answer:
a. (24, 2)
b. (8, 6)
Step-by-step explanation:
The orthocenter is the point where the altitudes of the triangle coincide. For an obtuse triangle, it will be outside the triangle. For a right triangle, it will be the right angle vertex. Of course, each altitude is a line through a vertex and perpendicular to the opposite side.
To make finding the orthocenter as easy as possible, we want to find a simple way to write an equation for a line through a point and perpendicular to the segment between two other points. For segment end points (x1, y1) and (x2, y2), the line between them will have slope (y2 -y1)/(x2 -x1). The perpendicular line has a slope that is the opposite reciprocal of this: -(x2 -x1)/(y2 -y1). Then the point-slope equation of the perpendicular through point (x3, y3) is ...
y -y3 = -(x2 -x1)/(y2 -y1)(x -x3)
This can be rearranged to something resembling standard form:
(x2 -1x)(x -x3) +(y2 -y1)(y -y3) = 0
(x2 -x1)x +(y2 -y1)y = (x2 -x1)x3 +(y2 -y1)y3 . . . . line through P3, ⊥ to P1P2
__
a.
For the first altitude, we choose P1 = (0, 0), P2 = (7, 3), and P3 = (3, 51). This gives ...
- (x2 -x1) = 7 -0 = 7
- (y2 -y1) = 3 -0 = 3
- altitude equation = 7x +3y = 7·3 +3·51 = 174
For the second altitude, we choose P1 = (0, 0), P2 = (3, 51), and P3 = (7, 3). This gives ...
- (x2 -x1) = 3 -0 = 3
- (y2 -y1) = 51 -0 = 51
- altitude equation = 3x +51y = 3·7 +51·3 = 174
Finding the point of intersection of these two equations can be done an of a number of ways. Cramer's Rule provides a quick solution:
x = ((3)(-174) -(51)(-174))/(7·51 -3·3) = 8352/348 = 24
y = ((-174)(3) -(-174)(7))/348 = 696/348 = 2
The orthocenter for the triangle is (24, 2).
__
b.
We can use the method just described, or we can verify the triangle is a right triangle, as it appears to be when the points are plotted.
The slope of P1P2 is (6 -3)/(8 -5) = 3/3/ = 1.
The slope of P2P3 is (10 -6)/(4 -8) = 4/-4 = -1
These slopes have a product of -1, so the segments are perpendicular. The vertex P2 has the right angle, hence is the orthocenter.
The orthocenter for the triangle is (8, 6).
