What is the orthocenter of a triangle with corners at (3 ,2 ), (4 ,5 ), and (2 ,7 )#?

1 Answer
Binayaka C.
Nov 4, 2017

Orthocenter of the triangle is at ( 5.5,6.5)

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 = (3,2) , B(4,5) , C(2,7) . 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= (7-5)/(2-4)= -1
Slope of perpendicular AD is m_2= 1 (m_1*m_2=-1)

Equation of line AD passing through A(3,2) is y-2= 1(x-3) or
y-2 = x-3 or x-y=1 (1)

Slope of AB is m_1= (5-2)/(4-3) =3
Slope of perpendicular CF is m_2= -1/3 (m_1*m_2=-1)

Equation of line CF passing through C(2,7) is y-7= -1/3(x-2) or
y-7 = -1/3 x+2/3 or 1/3x+y = 7+2/3 or 1/3x+y = 23/3 or
x+3y=23 (2)

Solving equation(1) and (2) we get their intersection point , which is the orthocenter.

x-y=1 (1) ; x+3y=23(2) Subtracting (1) from (2) we get,
4y=22 :. y=5.5 ; x = y+1=6.5

Orthocenter of the triangle is at ( 5.5,6.5) [Ans]

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