Question

Find the orthocenter for the triangle described by each set of vertices.

Answer 1

Answer:

The orthocentre of the given vertices ( 2 , -3.5)

Step-by-step explanation:

Step(i):-

The orthocentre is the intersecting point for all the altitudes of the triangle.

The point where the altitudes of a triangle meet is known as the orthocentre.

Given Points are K (3.-3), L (2,1), M (4,-3)

The Altitudes are perpendicular line from one side of the triangle to the opposite vertex

The altitudes are  MN , KO , LP

step(ii):-

   Slope of the line  

                          $KL = \frac{y_{2}-y_{1} }{x_{2}-x_{1} } = \frac{1-(-3)}{2-3} = -4$

The slope of MN =

The perpendicular slope of KL

                           = $\frac{-1}{m} = \frac{-1}{-4} = \frac{1}{4}$

The equation of the altitude

                                 $y - y_{1} = m( x-x_{1} )$

                                $y - (-3) = \frac{1}{4} ( x-4 )$

                               4y +12 = x -4

                                x - 4 y -16 = 0 ...(i)

Step(iii):-

 Slope of the line  

                          $LM = \frac{y_{2}-y_{1} }{x_{2}-x_{1} } = \frac{-3-1}{4-2} = -2$

The slope of KO =

The perpendicular slope of LM

                           = $\frac{-1}{m} = \frac{-1}{-2} = \frac{1}{2}$

The equation of the altitude

                                 $y - y_{1} = m( x-x_{1} )$

The equation of the line passing through the point K ( 3,-3) and slope

m = 1/2

                                $y - (-3) = \frac{1}{2} ( x-3 )$

                                2y +6 = x -3

                             x - 2y -9 =0 ....(ii)    

Solving equation (i) and (ii) , we get

  subtracting equation (i) and (ii) , we get

                   x - 4y -16 -( x-2y-9) =0

                       - 2y -7 =0

                      -2y = 7

                        y = - 3.5

Substitute y = -3.5 in equation x -4y-16=0

               x - 4( -3.5) - 16 =0

               x +14-16 =0

                x -2 =0

                  x = 2

The orthocentre of the given vertices ( 2 , -3.5)