What is the orthocenter of a triangle with corners at #(1, 3)#, #(6, 2)#, and #(5, 4)#?

Answer 1

#(x, y)=(47/9, 46/9)#

Let: A(1, 3), B(6, 2) and C(5, 4) be the vertices of triangle ABC:

Slope of a line through points: #(x_1, y_1), (x_2, y_2)#: #m=(y_2-y_1)/(x_2-x_1)#
Slope of AB: #=(2-3)/(6-1)=-1/5# Slope of perpendicular line is 5. Equation of the altitude from C to AB: #y-y_1=m(x-x_1)# =>#m=5, C(5,4)#: #y-4=5(x-5)# #y=5x-21#
Slope of BC: #=(4-2)/(5-6)=-2# Slope of perpendicular line is 1/2. Equation of the altitude from A to BC: #y-3=1/2(x-1)# #y=(1/2)x+5/2#
The intersection of the altitudes equating y's: #5x-21=(1/2)x+5/2# #10x-42=x+5# #9x=47# #x=47/9#
#y=5*47/9- 21# #y=46/9#
Thus the Orthocenter is at #(x, y)=(47/9, 46/9)#

To check the answer you can find the equation of altitude from B to AC and find the intersection of that with one of the other altitudes.

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Answer 2

The orthocenter of the triangle with vertices at (1, 3), (6, 2), and (5, 4) is located at the point (4.8, 2.6).

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Answer 3

To find the orthocenter of a triangle, you need to find the point where the three altitudes of the triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side. To find the altitude, you first find the slope of the line containing the side opposite the vertex and then use the negative reciprocal of that slope to find the slope of the altitude. Next, you use the point-slope form of a line equation to find the equation of the altitude. Then, you find the point of intersection of the three altitudes, which is 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.

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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