The orthocenter is the intersecting point for all the altitudes of the triangle. When given the three coordinates of a triangle, we can find equations for two of the altitudes, and then find where they intersect to get the orthocenter.
Let's call #color(red)((4,9)#, #color(blue)((7,4)#, and #color(green)((8,1)# coordinates #color(red)(A#,# color(blue)(B#, and #color(green)(C# respectively. We'll find equations for lines #color(crimson)(AB# and #color(cornflowerblue)(BC#. To find these equations, we'll need a point and a slope. (We'll use the point-slope formula).
Note: The slope of the altitude is perpendicular to the slope of the lines. The altitude will touch a line and the point that lies outside of the line.
First, let's tackle #color(crimson)(AB#:
Slope: #-1/({4-9}/{7-4})=3/5#
Point: #(8,1)#
Equation: #y-1=3/5(x-8)->color(crimson)(y=3/5(x-8)+1#
Then, let's find #color(cornflowerblue)(BC#:
Slope: #-1/({1-4}/{8-7})=1/3#
Point: #(4,9)#
Equation: #y-9=1/3(x-4)->color(cornflowerblue)(y=1/3(x-4)+9#
Now, we just set the equations equal to each other, and the solution would be the orthocenter.
#color(crimson)(3/5(x-8)+1)=color(cornflowerblue)(1/3(x-4)+9#
#(3x)/5-24/5+1=(x)/3-4/3+9#
#-24/5+1+4/3-9=(x)/3-(3x)/5#
#-72/15+15/15+20/15-135/15=(5x)/15-(9x)/15#
#-172/15=(-4x)/15#
#color(darkmagenta)(x=-172/15*-15/4=43#
Plug the #x#-value back into one of the original equations to get the y-coordinate.
#y=3/5(43-8)+1#
#y=3/5(35)+1#
#color(coral)(y=21+1=22#
Orthocenter: #(color(darkmagenta)(43),color(coral)(22))#
