# What is the orthocenter of a triangle with corners at #(1 ,4 )#, #(5 ,7 )#, and (2 ,3 )#?

Orthocenter is at

In order to find the orthocenter, we must derive two altitude linear equations from the three vertices that are provided.

An altitude equation can be obtained by taking one negative reciprocal of the slope from (1, 4) to (5, 7) and the point (2, 3).

Another altitude equation is given by the negative reciprocal of the slope from (2, 3) to (5, 7) and the point (1, 4).

Utilizing the first and second equations, solve the orthocenter

May God bless you all. I hope this explanation helps.

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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. An altitude is a line segment drawn from a vertex of the triangle perpendicular to the opposite side. Here are the steps to find the orthocenter:

- Find the slopes of the lines passing through each pair of vertices.
- Find the equations of the perpendicular bisectors for each side of the triangle.
- Find the point of intersection of these perpendicular bisectors, which is the orthocenter.

Alternatively, you can use the slopes of the sides of the triangle and the coordinates of the vertices to directly compute the coordinates of the orthocenter.

Let's go ahead with the alternative method:

Given the vertices: (A(1, 4)), (B(5, 7)), (C(2, 3)).

Step 1: Calculate the slopes of the sides of the triangle using the formula: (m = \frac{{y_2 - y_1}}{{x_2 - x_1}}).

The slopes of the sides are: (m_{AB} = \frac{{7 - 4}}{{5 - 1}} = \frac{3}{4}), (m_{BC} = \frac{{3 - 7}}{{2 - 5}} = \frac{-4}{3}), (m_{AC} = \frac{{3 - 4}}{{2 - 1}} = -1).

Step 2: Use the slope of each side and the coordinates of the vertices to find the equations of the altitudes (perpendicular lines passing through each vertex).

Step 3: Solve the system of equations formed by the equations of the altitudes to find the coordinates of the orthocenter.

By following these steps, you can find the orthocenter of the triangle with vertices at (1, 4), (5, 7), and (2, 3).

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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 orthocenter of a triangle with corners at #(4 ,1 )#, #(7 ,4 )#, and (2 ,8 )#?
- What is the orthocenter of a triangle with corners at #(1 ,3 )#, #(5 ,7 )#, and (2 ,3 )#?
- What is the orthocenter of a triangle with corners at #(9 ,5 )#, #(4 ,4 )#, and (8 ,6 )#?
- A triangle has corners A, B, and C located at #(5 ,6 )#, #(3 ,5 )#, and #(1 , 2 )#, respectively. What are the endpoints and length of the altitude going through corner C?
- Explain two attributes a parallelogram and a rectangle have in common ?

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