What is the orthocenter of a triangle with corners at #(4 ,9 )#, #(7 ,4 )#, and (8 ,1 )#?

Question

What is the orthocenter of a triangle with corners at #(4  ,9  )#, #(7 ,4   )#, and (8   ,1   )#?

Caroline Aucoin · Accepted Answer

Orthocenter: #(43,22)#

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))#

Caroline Aucoin · Answer

undefined The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. To find the orthocenter, you would first need to determine the equations of the altitudes and then find their point of intersection.

Since the altitudes are perpendicular to their respective opposite sides, you can use the slope-point form to find the equations of the altitudes. Once you have the equations, you can solve them simultaneously to find the coordinates of the orthocenter.

Alternatively, a quicker method is to use the fact that the orthocenter is the intersection of the three altitudes and apply the orthocenter formula, which involves the dot product and cross product of the triangle's edges. Using either method, you would eventually arrive at the coordinates of the orthocenter of the given triangle.