What is the orthocenter of a triangle with corners at $(6 ,3 )$ , $(4 ,5 )$ , and (2 ,9 )#?

Question

2770 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-07-02T10:17:42

The orthocenter of triangle is $(-14,-7)$

Let $triangle ABC$ be the triangle with corners at

$A(6,3) ,B(4,5) and C(2,9)$

Let $bar(AL) , bar(BM) and bar(CN)$ be the altitudes of sides

$bar(BC) ,bar(AC) ,and bar(AB)$ respectively.

Let $(x,y)$ be the intersection of three altitudes .

enter image source here

Slope of $bar(AB) =(5-3)/(4-6)=-1$

$bar(AB)_|_bar(CN)=>$ slope of $bar(CN)=1$ , $bar(CN)$ passes through $C(2,9)$

$:.$ The equn. of $bar(CN)$ is $:y-9=1(x-2)$

$i.e. color(red)(x-y=-7.....to (1)$

Slope of $bar(BC) =(9-5)/(2-4)=-2$

$bar(AL)_|_bar(BC)=>$ slope of $bar(AL)=1/2$ , $bar(AL)$ passes through $A(6,3)$

$:.$ The equn. of $bar(AL)$ is $:y-3=1/2(x-6)=>2y-6=x-6$

$i.e. color(red)(x=2y.....to (2)$

Subst. $x=2y$ into $(1)$ ,we get

$2y-y=-7=>color(blue)( y=-7$

From equn. $(2)$ we get

$x=2y=2(-7)=>color(blue)(x=-14$

Hence, the orthocenter of triangle is $(-14,-7)$

What is the orthocenter of a triangle with corners at (6 ,3 )(6 ,3 ), (4 ,5 )(4 ,5 ), and (2 ,9 )#?