What is the orthocenter of a triangle with corners at #(4 ,5 )#, #(8 ,3 )#, and #(5 ,9 )#?

Answer 1

The orthocenter is #=(8/3,13/3)#

Let the triangle #DeltaABC# be
#A=(4,5)#
#B=(8,3)#
#C=(5,9)#
The slope of the line #BC# is #=(9-3)/(5-8)=-6/3=-2#
The slope of the line perpendicular to #BC# is #=1/2#
The equation of the line through #A# and perpendicular to #BC# is
#y-5=1/2(x-4)#...................#(1)#
#2y=x-4+10=x+6#
The slope of the line #AB# is #=(3-5)/(8-4)=-2/4=-1/2#
The slope of the line perpendicular to #AB# is #=2#
The equation of the line through #C# and perpendicular to #AB# is
#y-9=2(x-5)#
#y-9=2x-10#
#y=2x-1#...................#(2)#
Solving for #x# and #y# in equations #(1)# and #(2)#
#4x-2=x+6#
#4x-x=6+2#
#3x=8#
#x=8/3#
#y=2x-1=2*8/3-1=13/3#
The orthocenter of the triangle is #=(8/3,13/3)#
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Answer 2

To find the orthocenter of a triangle, you need to determine the point where the three altitudes of the triangle intersect. An altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side. To find the orthocenter, you can follow these steps:

  1. Calculate the slopes of the lines containing the sides of the triangle using the given coordinates.
  2. Determine the slopes of the perpendicular lines (altitudes) passing through each vertex.
  3. Use the point-slope form to find the equations of the altitudes.
  4. Find the intersection point of the altitudes, which will be the orthocenter of the triangle.

Alternatively, you can use the properties of perpendicular lines to find the equations of the altitudes more directly. Once you have the equations of the altitudes, solve the system of equations to find the coordinates of the orthocenter.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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