What is the orthocenter of a triangle with corners at #(2 ,6 )#, #(9 ,1 )#, and (5 ,3 )#?

Answer 1

The Orthocenter is #(-10,-18)#

The Orthocenter of a triangle is the point of intersection of the 3 altitudes of the triangle.

The slope of the line segment from point #(2,6)# to #(9,1) # is:
#m_1 = (1-6)/(9-2)#
#m_1 = -5/7#

The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:

#p_1 = -1/m_1#
#p_1 = -1/(-5/7)#
#p_1 = 7/5#
The altitude must pass through point #(5,3)#

We can use the point-slope form for the equation of a line to write the equation for the altitude:

#y = 7/5(x-5)+3#

Simplify a bit:

#y = 7/5x-4" [1]"#
The slope of the line segment from point #(2,6)# to #(5,3) # is:
#m_2 = (3-6)/(5-2)#
#m_2 = -3/3#
#m_2 = -1#

The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:

#p_2 = -1/m_2#
#p_2 = -1/(-1)#
#p_2 = 1#
The altitude must pass through point #(9,1)#

We can use the point-slope form for the equation of a line to write the equation for the altitude:

#y = 1(x-9)+1#

Simplify a bit:

#y = x-8" [2]"#

We could repeat this process for the third altitude but we have already enough information to determine the intersection point.

Set the right side of equation [1] equal to the right side of equation [2]:

#7/5x-4 = x-8#

Solve for the x coordinate of intersection:

#2/5x =-4#
#x = -10#

To find the value of y, substitute -10 for x into equation [2]:

#y = -10 - 8#
#y = -18#
The Orthocenter is #(-10,-18)#
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Answer 2

To find the orthocenter of a triangle, you need to find the point where all three altitudes of the triangle intersect.

  1. Find Slopes of the Sides: Calculate the slopes of the sides formed by the given points.

  2. Find the Slopes of the Altitudes: Perpendicular slopes to the sides of the triangle can be found by taking negative reciprocals of the slopes of the sides.

  3. Find Equations of Altitudes: Use the point-slope form with each vertex to find the equations of the altitudes.

  4. Find Intersection Point: Solve the system of equations formed by the altitudes to find the orthocenter.

  5. Check Validity: Ensure that the intersection point lies within the triangle to confirm it as the orthocenter.

This process can be a bit lengthy, but it provides the exact location 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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