What is the orthocenter of a triangle with corners at $(9 ,7 )$ , $(4 ,1 )$ , and (8 ,2 )#?

Question

2005 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-06-22T12:14:38

The orthocenter of the triangle is $=(206/19,-7/19)$

Let the triangle $DeltaABC$ be

$A=(9,7)$

$B=(4,1)$

$C=(8,2)$

The slope of the line $BC$ is $=(2-1)/(8-4)=1/4$

The slope of the line perpendicular to $BC$ is $=-4$

The equation of the line through $A$ and perpendicular to $BC$ is

$y-7=-4(x-9)$ ................... $(1)$

$y=-4x+36+7=-4x+43$

The slope of the line $AB$ is $=(1-7)/(4-9)=-6/-5=6/5$

The slope of the line perpendicular to $AB$ is $=-5/6$

The equation of the line through $C$ and perpendicular to $AB$ is

$y-2=-5/6(x-8)$

$y-2=-5/6x+20/3$

$y+5/6x=20/3+2=26/3$ ................... $(2)$

Solving for $x$ and $y$ in equations $(1)$ and $(2)$

$-4x+43=26/3-5/6x$

$4x-5/6x=43-26/3$

$19/6x=103/3$

$x=206/19$

$y=26/3-5/6x=26/3-5/6*206/19=26/3-1030/114=-42/114=-7/19$

The orthocenter of the triangle is $=(206/19,-7/19)$

What is the orthocenter of a triangle with corners at (9 ,7 )(9 ,7 ), (4 ,1 )(4 ,1 ), and (8 ,2 )#?