What is the orthocenter of a triangle with corners at #(5 ,7 )#, #(2 ,3 )#, and (4 ,5 )#?

Answer 1

Orthocenter of the triangle is at #( 16,-4) #

Orthocenter is the point where the three "altitudes" of a triangle

meet. An "altitude" is a line that goes through a vertex (corner

point) and is perpendicular to the opposite side.

#A = (5,7) , B(2,3) , C(4,5) # . Let #AD# be the altitude from #A#
on #BC# and #CF# be the altitude from #C# on #AB# they meet at
point #O# , the orthocenter.
Slope of line #BC# is #m_1= (5-3)/(4-2)= 1#
Slope of perpendicular #AD# is #m_2= -1 (m_1*m_2=-1) #
Equation of line #AD# passing through #A(5,7)# is
#y-7= -1(x-5) or y-7 =-x+5 or x+y =12 ; (1)#
Slope of line #AB# is #m_1= (3-7)/(2-5)=4/3#
Slope of perpendicular #CF# is #m_2= -3/4 (m_1*m_2=-1) #
Equation of line #CF# passing through
#C(4,5)# is #y-5= -3/4(x-4) or 4 y - 20 = -3 x +12 # or
# 3 x+4 y =32; (2)# Solving equation(1) and (2) we get their

intersection point , which is the orthocenter. Multiplying

equation (1) by #3# we get, #3 x+3 y =36 ; (3)# Subtracting

equation (3) from equation (2) we get,

#y = -4 :. x=12-y= 12+4=16 :. (x,y) = (16 , -4)#
Hence Orthocenter of the triangle is at #( 16,-4) # [Ans]
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Answer 2

To find the orthocenter of a triangle, you need to find the point where the altitudes of the triangle intersect. The altitudes are the perpendicular lines from each vertex to the opposite side.

  1. First, find the slopes of the lines containing the sides of the triangle.
  2. Then, find the equations of the perpendicular bisectors of each side (lines perpendicular to the sides passing through the midpoints of the sides).
  3. The point of intersection of these perpendicular bisectors is the circumcenter.
  4. Finally, find the equation of the altitude from one vertex to the opposite side. The point where all three altitudes intersect is the orthocenter.

Alternatively, you can use the orthocenter formula directly, which involves the coordinates of the vertices.

The orthocenter of the triangle with corners at (5,7), (2,3), and (4,5) is (4, 5).

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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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