Find the coordinates of the orthocenter of a triangle with vertices at each set of points on a coordinate plane.
a. (0,0), (7,3), (3,51)
b. (5,3), (8,6), (4,10)
.
a. The orthocenter is .
(Type an ordered pair. Simplify your answer.)

b. The orthocenter is.
(Type an ordered pair. Simplify your answer.)

Respuesta :

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).

Ver imagen sqdancefan
Ver imagen sqdancefan
ACCESS MORE
EDU ACCESS