What is the orthocenter of a triangle with corners at #(2 ,6 )#, #(9 ,1 )#, and (5 ,3 )#?
The Orthocenter is
The Orthocenter of a triangle is the point of intersection of the 3 altitudes of the triangle.
The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:
We can use the pointslope form for the equation of a line to write the equation for the altitude:
Simplify a bit:
The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:
We can use the pointslope form for the equation of a line to write the equation for the altitude:
Simplify a bit:
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]:
Solve for the x coordinate of intersection:
To find the value of y, substitute 10 for x into equation [2]:
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To find the orthocenter of a triangle, you need to find the point where all three altitudes of the triangle intersect.

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

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.

Find Equations of Altitudes: Use the pointslope form with each vertex to find the equations of the altitudes.

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

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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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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 A line segment is bisected by a line with the equation # 4 y  6 x = 8 #. If one end of the line segment is at #( 8 , 3 )#, where is the other end?
 A triangle has corners A, B, and C located at #(4 ,2 )#, #(1 ,3 )#, and #(6 ,5 )#, respectively. What are the endpoints and length of the altitude going through corner C?
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