What is the orthocenter of a triangle with corners at #(2 ,0 )#, #(3 ,4 )#, and (6 ,3 )#?

Answer 1

The orthocenter of the triangle is : # (42/13,48/13)#

Let #triangleABC# be the triangle with corners at

#A(2,0), B(3,4) and C(6,3)#.

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 .

#diamond#Slope of #bar(AB)#=#(4-0)/(3-2)#=#4=>#slope of #bar(CN)#=#-1/4[because#altitudes]

Now, #bar(CN)# passes through #C(6,3)#

#:.# Equn. of #bar(CN)# is: #y-3=-1/4(x-6)#

#i.e. color(red)(x+4y=18...to(1)#

#diamond#Slope of #bar(BC)#=#(3-4)/(6-3)#=#-1/3=>#slope of #bar(AL)=3[because#altitudes]

Now, #bar(AL)# passes through #A(2,0)#

#:.# Equn. of #bar(AL)# is: #y-0=3(x-2)#

#i.e. color(red)(3x-y=6...to(2)#

#=>color(red)(y=3x-6...to(3)#

Putting ,#y=3x-6# into #(1)# we get

#x+4(3x-6)=18=>x+12x-24=18#

#=>13x=42#

#=>color(blue)(x=42/13#

From #(3)# we get,

#y=3(42/13)-6=(126-78)/13#

#=>color(blue)(y=48/13#

Hence, **the orthocenter of the triangle is :

** # (42/13,48/13)~~(3.23,3.69)#

Please see the graph.

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Answer 2

To find the orthocenter of a triangle, you need to find the point where the three altitudes of the triangle intersect. An altitude is a line segment from a vertex of the triangle perpendicular to the opposite side.

First, find the slopes of the lines containing the sides of the triangle. Then, determine the slopes of the lines perpendicular to these sides (these are the altitudes). Next, find the equations of these perpendicular lines. Finally, solve the system of equations formed by these lines to find their point of intersection, which is the orthocenter.

Given the vertices (2, 0), (3, 4), and (6, 3), calculate the slopes of the lines passing through these points. Then, find the perpendicular slopes. Using these perpendicular slopes and the corresponding vertices, find the equations of the altitudes. Solve the system of equations formed by these lines to find their intersection point, which is 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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