# What is the orthocenter of a triangle with corners at #(5 ,2 )#, #(3 ,3 )#, and (7 ,9 )#?

The orthocenter is a point where the altitudes of a triangle meet. In order to find this point we must find two of the three lines and their point of intersection. We do not need to find all three lines, since the intersection of two of these will uniquely define a point in a two dimensional space.

Labelling vertices:

We need to find two lines that are perpendicular to two of the sides of the triangle. We first find the slopes of two sides.

The line perpendicular to AB passes through C. The gradient of this will be the negative reciprocal of the gradient of AB. Using point slope form:

The line perpendicular to AC passes through B. Gradient negetive reciprocal of AC:

We now find the point of intersection of these two lines. Solving simultaneously:

So the orthocenter is at:

PLOT:

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To find the orthocenter of a triangle, you first need to determine the altitude of each side of the triangle. The altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. The point where all three altitudes intersect is the orthocenter.

Given the coordinates of the triangle's vertices, you can use the slope formula to find the slopes of the lines containing the sides of the triangle. Then, using the perpendicular slope property, you can find the slopes of the lines perpendicular to the sides, which represent the altitudes. Finally, using the point-slope form of a line, you can find the equations of the altitudes and solve them simultaneously to find the orthocenter.

Alternatively, you can use a geometric approach by drawing the perpendicular bisectors of the sides of the triangle and finding their point of intersection, which is also the orthocenter.

After applying either method, the coordinates of the orthocenter of the triangle with corners at (5, 2), (3, 3), and (7, 9) will be provided.

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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 #(4 ,3 )#, #(7 ,4 )#, and #(6 ,5 )#, 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 - x = 3 #. If one end of the line segment is at #( 8 , 3 )#, where is the other end?
- A line segment is bisected by a line with the equation # 3 y - 7 x = 3 #. If one end of the line segment is at #(7 ,8 )#, where is the other end?
- What is the orthocenter of a triangle with corners at #(6 ,2 )#, #(4 ,7 )#, and (8 ,3 )#?
- A triangle has corners A, B, and C located at #(1 ,3 )#, #(7 ,9 )#, and #(5 , 8 )#, respectively. What are the endpoints and length of the altitude going through corner C?

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