# What is the orthocenter of a triangle with corners at #(4 ,9 )#, #(3 ,4 )#, and (5 ,1 )#?

The orthocenter of the triangle is

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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 find the intersection point of the altitudes.

- Find the slopes of the lines passing through each pair of vertices to determine the slopes of the altitudes.
- Use the point-slope form to write the equations of the altitudes.
- Find the intersection point of any two altitude lines to determine the orthocenter.

Given the coordinates of the vertices: A (4, 9) B (3, 4) C (5, 1)

- Find the slopes of the lines AB, BC, and CA using the formula: (m = \frac{{y_2 - y_1}}{{x_2 - x_1}}).

(m_{AB} = \frac{{4 - 9}}{{3 - 4}} = 5)

(m_{BC} = \frac{{1 - 4}}{{5 - 3}} = -\frac{3}{2})

(m_{CA} = \frac{{1 - 9}}{{5 - 4}} = -8)

- Use the point-slope form (y - y_1 = m(x - x_1)) to write the equations of the altitudes passing through each vertex.

Equation of the altitude from A: (y - 9 = 5(x - 4)) (y - 9 = 5x - 20) (y = 5x - 11)

Equation of the altitude from B: (y - 4 = -\frac{3}{2}(x - 3)) (y - 4 = -\frac{3}{2}x + \frac{9}{2}) (y = -\frac{3}{2}x + \frac{17}{2})

Equation of the altitude from C: (y - 1 = -8(x - 5)) (y - 1 = -8x + 40) (y = -8x + 41)

- Now, find the intersection points of any two altitude lines. For example, let's find the intersection of altitudes from A and B:

(5x - 11 = -\frac{3}{2}x + \frac{17}{2})

Solve for (x):

(5x + \frac{3}{2}x = \frac{17}{2} + 11)

(x = \frac{29}{7})

Substitute (x) back into one of the altitude equations to find (y):

(y = 5 \times \frac{29}{7} - 11)

(y = \frac{29}{7})

So, the intersection point of the altitudes from A and B is (\left(\frac{29}{7}, \frac{29}{7}\right)).

Similarly, find the intersection points of other pairs of altitude lines. The point where all three altitudes intersect is 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.

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