EXAMPLE 12 The vertices of a triangle are A (10,4), B(-4,9)and C (-2,-1). Find the equation of its
altitudes. Also, find its orthocentre.
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Answer:
The vertices of a triangle are A (10,4), B (-4,9) & (-2, -1). What are the coordinates of its orthocenter?
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Let perpendiculars drawn from A to BC is AE and from B to AC is BF , thus
AE and BF meet at point P which is the orthocenter of ∆ABC.
Slope of BC=(-1–9)/(-2+4)= -5
Thus ,slope of AE = 1/5.
Equation of AE is:-
y-4= 1/5.(x-10)
or. x-10=5y-20
or. x-5y = -10……………..(1)
Slope of AC =(-1–4)/(-2–10)= 5/12.
Slope of BF =-12/5.
Equation of BF is:-
y-9=-12/5.(x+4).
or 5y-45=-12x-48
or. 12x+5y= -3………………..(2)
from eqn (1) and (2).
x/(15+50)=y/(-120+3)= -1/(5+60)
x/65=y/-117 =-1/65.
x=-65/65= -1.
y= 117/65 = 9/5.
Coordinates of the orthocenter = (-1 ,9/5).