What is the orthocenter of a triangle with corners at $(9 ,7 )$ , $(2 ,4 )$ , and (8 ,6 )#?

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Answer 1 · 2018-07-30T06:48:56

The orthocenter of triangle is $(14,-8)$

Let $triangleABC " be the triangle with corners at"$

$A(9,7), B(2,4) and C(8,6)$

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 .

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Slope of $bar(AB) =(7-4)/(9-2)=3/7$

$bar(AB)_|_bar(CN)=>$ slope of $bar(CN)=-7/3$ ,

$bar(CN)$ passes through $C(8,6)$

$:.$ The equn. of $bar(CN)$ is $:y-6=-7/3(x-8)$

$3y-18=-7x+56$

$i.e. color(red)(7x+3y=74.....to (1)$

Slope of $bar(BC) =(6-4)/(8-2)=2/6=1/3$

$bar(AL)_|_bar(BC)=>$ slope of $bar(AL)=-3$ , $bar(AL)$ passes through $A(9,7)$

$:.$ The equn. of $bar(AL)$ is $:y-7=-3(x-9)=>y-7=-3x+27$

$=>3x+y=34$

$i.e. color(red)(y=34-3x.....to (2)$

Subst. $color(red)(y=34-3x$ into $(1)$ ,we get

$7x+3(34-3x)=74=>7x+102-9x$ = $74=>-2x=-28$
$=>color(blue)( x=14$

From equn. $(2)$ we get

$y=34-3(14)=34-42=>color(blue)(y=-8$

Hence, the orthocenter of triangle is $(14,-8)$

What is the orthocenter of a triangle with corners at (9 ,7 )(9 ,7 ), (2 ,4 )(2 ,4 ), and (8 ,6 )#?