# What is the orthocenter of a triangle with corners at #(4 ,7 )#, #(9 ,2 )#, and (5 ,6 )#?

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The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. To find the orthocenter, you need to determine the equations of the altitudes and solve for their point of intersection. Given the coordinates of the triangle's vertices, you can calculate the slopes of the sides, find the perpendicular bisectors of those sides to get the equations of the altitudes, and then solve for their point of intersection. Alternatively, you can use trigonometric methods or the properties of perpendicular lines to find the orthocenter. Once you have the equations of the altitudes or the necessary angles and distances, you can calculate the coordinates of the orthocenter.

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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.

- What is the centroid of a triangle with corners at #(4 , 1 )#, #(3 , 2 )#, and #(5 , 0 )#?
- A line segment is bisected by a line with the equation # 4 y + x = 8 #. If one end of the line segment is at #(5 ,2 )#, where is the other end?
- A line segment is bisected by a line with the equation # - 2 y - 5 x = 2 #. If one end of the line segment is at #( 8 , 7 )#, where is the other end?
- What is the orthocenter of a triangle with corners at #(5 ,2 )#, #(3 ,7 )#, and (4 ,9 )#?
- A line segment is bisected by a line with the equation # 5 y + 2 x = 1 #. If one end of the line segment is at #(6 ,4 )#, where is the other end?

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