What is the orthocenter of a triangle with corners at #(4 ,3 )#, #(9 ,5 )#, and (8 ,6 )#?
1 Answer
Using the corners of the triangle, we can get the equation of each perpendicular; using which, we can find their meeting point
Explanation:
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The rules we're gonna use are:
The given triangle has corners A,B, and C in the order given above.
The slope of a line that passes through#(x_1,y_1) , (x_2,y_2)# has slope =#(y_1-y_2)/(x_1-x_2)#
Line A which is perpendicular to line B has#"slope"_A=-1/"slope"_B # -
The slope of:
Line AB =#2/5#
Line BC =#-1#
Line AC =#3/4# -
The slope of the line perpendicular to each side:
Line AB =#-5/2#
Line BC =#1#
Line AC =#-4/3# -
Now you can find the equation of each perpendicular bisector passing through the opposite corner. For example, the line perpendicular to AB passing through C. They are, in the order used above:
#y-6=-5/2(x-8)#
#y-3=x-4#
#y-5=-4/3(x-9)# -
If you solve any two of these 3, you'll get their meeting point-the orthocenter. Which is
#(54/7,47/7)# .
