# What is the orthocenter of a triangle with corners at #(3 ,6 )#, #(4 ,2 )#, and (5 ,7 )#?

Orthocenter of the triangle

Slope of

Slope of

Equation of AD is

Slope of

Slope of

Equation of CF is

Solving Eqns (1) & (2), we get the orthocenter

Solving the two equations,

Coordinates of orthocenter

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To find the orthocenter of a triangle, you need to find the point of intersection of the altitudes of the triangle. An altitude is a line segment from a vertex perpendicular to the opposite side.

First, find the slopes of the lines containing the sides of the triangle:

Slope of the line passing through (3, 6) and (4, 2): m1 = (2 - 6) / (4 - 3) = -4 / 1 = -4

Slope of the line passing through (3, 6) and (5, 7): m2 = (7 - 6) / (5 - 3) = 1 / 2

Slope of the line passing through (4, 2) and (5, 7): m3 = (7 - 2) / (5 - 4) = 5 / 1 = 5

The negative reciprocal of the slope of a line is perpendicular to it.

So, the slopes of the altitudes are:

- For the altitude from (3, 6): m1_perpendicular = -1 / m1 = -1 / -4 = 1/4
- For the altitude from (4, 2): m2_perpendicular = -1 / m2 = -1 / (1/2) = -2
- For the altitude from (5, 7): m3_perpendicular = -1 / m3 = -1 / 5

Now, we have the slopes of the altitudes and the coordinates of the vertices. Using the point-slope form of a line, we can find the equations of the altitudes.

- For the altitude from (3, 6): y - 6 = (1/4)(x - 3)
- For the altitude from (4, 2): y - 2 = -2(x - 4)
- For the altitude from (5, 7): y - 7 = (-1/5)(x - 5)

Now, solve these equations to find the points of intersection, which will give you the orthocenter of the triangle.

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The orthocenter of a triangle is the point of intersection of its three altitudes. To find the orthocenter of a triangle with vertices at (3, 6), (4, 2), and (5, 7):

- Calculate the slopes of the three sides of the triangle.
- Find the slopes of the perpendicular lines (altitudes) passing through each vertex.
- Use the point-slope form to write the equations of these altitudes.
- Solve the system of equations to find the point of intersection, which is the orthocenter of the triangle.

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