What is the orthocenter of a triangle with corners at (6 ,3 ), (2 ,4 ), and (7 ,9 )#?

1 Answer
Binayaka C.
Jan 20, 2018

Orthocenter of the triangle is at (5.6,3.4)

Explanation:

Orthocenter is the point where the three "altitudes" of a triangle meet. An "altitude" is a line that goes through a vertex (corner point) and is at right angles to the opposite side.

A = (6,3) , B(2,4) , C(7,9) . Let AD be the altitude from A on BC and CF be the altitude from C on AB they meet at point O , the orthocenter.

Slope of BC is m_1= (9-4)/(7-2)=5/5= 1

Slope of perpendicular AD is m_2= -1 (m_1*m_2=-1)

Equation of line AD passing through A(6,3) is

y-3= -1(x-6)or y-3 = -x+6 or x +y = 9 (1)

Slope of AB is m_1= (4-3)/(2-6)= -1/4

Slope of perpendicular CF is m_2= -1/(-1/4)=4

Equation of line CF passing through C(7,9) is

y-9= 4(x-7) or y-9 = 4x-28 or 4x-y=19 (2)

Solving equation(1) and (2) we get their intersection point , which

is the orthocenter. Adding equation(1) and (2) we get,

5x=28 or x = 28/5=5.6 and y= 9-x =9-5.6 =3.4

Orthocenter of the triangle is at (5.6,3.4) [Ans]

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