Question

Find the co-ordinates of the orthocenter of
the triangle whose vertices are A(2,-2),
B(1,1) and C(-1,0).​

Answer 1

Answer :

Here, in the given figure AR and BD

are the altitudes and when they meet at a point then their point of intersection is called orthocenter of that triangle .

(Slope of AR)(Slope of BC) = -1

_________________ {perpendicular to each other } __(1)

Now,lets find out the slope of BC

B (1,1) , C (-1,0)

= { (y2-y1)/(x2-x1)}

= { (0-1)/(-1-1)} = -1/-2 = 1/2

A (2,-2) R (x,y)

from_ (1) :

(Slope of AR)(Slope of BC) = -1

(slope of AR) = -1/(slope of BC)

=-1 /(1/2)

= -2

find the equation of the altitude AR by slope point form

• slope point form :

(y - y1) =m (x - x1)

coordinates of A(2,-2) = (x1,y1)

=)(y -(-2))=-2(x-2)

= y+2 = -2x+4 , 2x+y-2 = 0 ___(i)

Now, let's find out the equation of altitude BD

BD is perpendicular to side of triangle AC

A (2,-2) , C (-1,0)

therefore, the slope of AC

= {(y2-y1)/(x2-x1)}

={(0-(-2))/(-1-2)}= 2/-3

(Slope of BD)(Slope of AC)= -1

___{perpendicular to each other }

(Slope of BD)(2/-3) = -1

slope of BD = 3/2

By slope point form the equation of altitude BD is ,B(1,1)= (x1,y1), D(x,y)

(y-y1)=m(x-x1) ,(y-1)=3/2(x-1)

2(y-1)=3(x-1) , 2y-2=3x-3

3x-2y = 1 __________(ii)

From equation (i) and (ii)

2x + y = 2 and 3x - 2y = 1

multiply 2 to the equation 2x+y=2

4x + 2y = 4 ________(iii)

take (-) on both the sides of equation 3x - 2y = 1,

therefore the equation becomes

-(3x-2y) =-1 , -3x+2y=-1 _____(iv)

subtract equation (iv) from equation (iii)

-3x + 2y = -1

4x + 2y = 4

- - -

______________

-7x = -5

x = 5/7

Substitute the value of x in equation iii

4x + 2y = 4

4(5/7) + 2y = 4

20+14y = 28

14y = 28 - 20 , y = 8/14 = 4/7

Hence the co-ordinates of the orthocenter are (5/7 , 4,7 )