What is the orthocenter of a triangle with corners at #(6 ,3 )#, #(4 ,5 )#, and (2 ,9 )#?

Answer 1

The orthocenter of triangle is #(-14,-7)#

Let #triangle ABC# be the triangle with corners at

#A(6,3) ,B(4,5) and C(2,9)#

Let #bar(AL) , bar(BM) and bar(CN) # be the altitudes of sides

#bar(BC) ,bar(AC) ,and bar(AB)# respectively.

Let #(x,y)# be the intersection of three altitudes .

Slope of #bar(AB) =(5-3)/(4-6)=-1#

#bar(AB)_|_bar(CN)=>#slope of # bar(CN)=1# , # bar(CN)# passes through #C(2,9)#

#:.#The equn. of #bar(CN)# is #:y-9=1(x-2)#

#i.e. color(red)(x-y=-7.....to (1)#

Slope of #bar(BC) =(9-5)/(2-4)=-2#

#bar(AL)_|_bar(BC)=>#slope of # bar(AL)=1/2# , # bar(AL)# passes through #A(6,3)#

#:.#The equn. of #bar(AL)# is #:y-3=1/2(x-6)=>2y-6=x-6#

#i.e. color(red)(x=2y.....to (2)#

Subst. #x=2y# into #(1)# ,we get

#2y-y=-7=>color(blue)( y=-7#

From equn.#(2)# we get

#x=2y=2(-7)=>color(blue)(x=-14#

Hence, the orthocenter of triangle is #(-14,-7)#

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Answer 2

To find the orthocenter of a triangle, you need to find the intersection point of the altitudes of the triangle. The altitude of a triangle is a line segment perpendicular to a side of the triangle, passing through the opposite vertex.

  1. Find the slopes of the sides of the triangle.
  2. Use the point-slope form to find the equations of the perpendicular bisectors of each side (these will be the altitudes).
  3. Find the intersection point of any two perpendicular bisectors to get the orthocenter.

For the given triangle with corners at (6, 3), (4, 5), and (2, 9):

  1. The slopes of the sides are:

    • Slope of side passing through (6, 3) and (4, 5) = (5-3) / (4-6) = 2/-2 = -1
    • Slope of side passing through (4, 5) and (2, 9) = (9-5) / (2-4) = 4/-2 = -2
    • Slope of side passing through (2, 9) and (6, 3) = (3-9) / (6-2) = -6/4 = -3/2
  2. Using the point-slope form (y - y1) = m(x - x1), we find the equations of the perpendicular bisectors:

    • For the side passing through (6, 3) and (4, 5), the midpoint is ((6+4)/2, (3+5)/2) = (5, 4). The slope of the perpendicular bisector is the negative reciprocal of -1, which is 1. Thus, the equation of the perpendicular bisector is y - 4 = 1(x - 5).
    • Similarly, for the other sides, you'll find the equations of the perpendicular bisectors.
  3. Solve any two equations of the perpendicular bisectors to find the orthocenter, which is the intersection point.

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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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