What is the orthocenter of a triangle with corners at #(5 ,2 )#, #(3 ,7 )#, and (4 ,9 )#?

Question

What is the orthocenter of a triangle with corners at #(5  ,2  )#, #(3  ,7   )#, and (4   ,9   )#?

Savannah Aguilar · Accepted Answer

#(-29/9, 55/9)#

Find the orthocenter of the triangle with vertices of #(5,2), (3,7),(4,9)#.

I will name the triangle #DeltaABC# with #A=(5,2)#, #B=(3,7)# and #C=(4,9)#

The orthocenter is the intersection of the altitudes of a triangle.

An altitude is a line segment that goes through a vertex of a triangle and is perpendicular to the opposite side.

If you find the intersection of any two of the three altitudes, this is the orthocenter because the third altitude will also intersect the others at this point.

To find the intersection of two altitudes, you must first find the equations of the two lines that represent the altitudes and then solve them in a system of equations to find their intersection.

First we will find the slope of the line segment between #A and B# using the slope formula #m=frac{y_2-y_1}{x_2-x_1}#

#m_(AB)=frac{7-2}{3-5}=-5/2#

The slope a line perpendicular to this line segment is the opposite sign reciprocral of #-5/2#, which is #2/5#.

Using the point slope formula #y-y_1=m(x-x_1)# we can find the equation of altitude from vertex #C# to side #AB#.

#y-9=2/5(x-4)#

#y-9=2/5 x -8/5#

#-2/5x+y=37/5color(white)(aaa)# or

#y=2/5 x +37/5#

To find the equation of a second altitude, find the slope of one of the other sides of the triangle. Let's choose BC.

#m_(BC)=frac{9-7}{4-3}=2/1=2#

The perpendicular slope is #-1/2#.

To find the equation of the altitude from vertex #A# to side #BC#, again use the point slope formula.

#y-2=-1/2(x-5)#

#y-2=-1/2x +5/2#

#1/2 x +y= 9/2#

The system of equations is

#color(white)(a^2)1/2 x +y=9/2# #-2/5x+y=37/5#

Solving this system yields #(-29/9, 55/9)#