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

Answer 1

Sophia Alfred

orthocenter #(79/11, 5/11)#

First, solve the altitude equations, and then the intersection of those equations.

through point-slope modeling

#y-2=-1/((7-3)/(5-4))(x-1)" "#equation of the altitude thru (1,2)

#y-3=-1/((7-2)/(5-1))(x-4)" "#equation of the altitude thru (4,3)

Simplifying these equations we have #x+4y=9# #4x+5y=31#

Concurrent resolution leads to

#x=79/11# and #y=5/11#

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

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Answer 2

Theodore Abad

The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. To find the orthocenter of a triangle with given vertices, you can follow these steps:

Find the slopes of the lines passing through each pair of vertices.
Determine the slopes of the altitudes, which are perpendicular to the sides of the triangle.
Use the point-slope form to find the equations of the altitudes.
Solve the system of equations formed by the altitudes to find the coordinates of the orthocenter.

Alternatively, you can use the properties of perpendicular bisectors and circumcenters to find the orthocenter.

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Answer from HIX Tutor

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.