# What is the orthocenter of a triangle with corners at #(9 , 5 )#, #(3 , 8 )#, and #(5 ,6 )#?

Steps: (1) find the slopes of 2 sides, (2) find the slopes of the lines perpendicular to those sides, (3) find the equations of the lines with those slopes that pass through the opposite vertices, (4) find the point where those lines intersect, which is the orthocenter, in this case

The slopes (gradients) of two of a triangle's sides and the equations of the lines perpendicular to those sides are what we need to find the orthocenter of the triangle.

The equations of the lines perpendicular to the sides that pass through the opposite angle can be found using those slopes in addition to the coordinates of the point opposite the relevant side; these lines are known as the "altitudes" for the sides.

The orthocenter is the point where the altitudes of two sides cross; the altitude of the third side would also pass through this point.

To make it simpler to refer to our points, let's label them:

Use the formula to determine the slope:

By entering the coordinates of Point C (opposite AB) and Point A (opposite BC) into the equation, we can now find the equations for their respective altitudes.

The altitude at Point C is:

In a similar vein, Point A:

We just need to locate the intersection of these two lines, which we can equate to each other to find the orthocenter:

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To find the orthocenter of a triangle, you first need to determine the perpendicular bisectors of each side of the triangle. Then, the intersection point of these perpendicular bisectors is the orthocenter.

For the given triangle with vertices at (9, 5), (3, 8), and (5, 6), you can calculate the slopes of the sides, find the midpoints of the sides, and then determine the perpendicular bisectors' equations. Finally, find the intersection point of these bisectors to get the orthocenter.

Alternatively, you can use the formula for the orthocenter, which involves the slopes of the sides and the vertices of the triangle. Once you have the slopes and the vertices, you can substitute them into the formula to find 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.

- A triangle has corners A, B, and C located at #(1 ,6 )#, #(9 ,3 )#, and #(2 ,2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- A line segment is bisected by a line with the equation # - 6 y + 5 x = 4 #. If one end of the line segment is at #( 2 , 5 )#, where is the other end?
- A line segment is bisected by a line with the equation # 2 y + x = 7 #. If one end of the line segment is at #( 5 , 3 )#, where is the other end?
- A triangle has corners at #(5 , 4 )#, ( 7, 1 )#, and #( 1, 3 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?
- What is the centroid of a triangle with corners at #(2 , 5 )#, #(4 ,7 )#, and #(2 , 7 )#?

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