Question

find the orthocenter of triangle (5,1) ,(1,2),(2,3)​

Answer 1

Answer:

Let A(5,−1) and B(−2,3) and orthocentre is O(0,0)

∴ Slope of altitude coming from A

=

5−0

−1−0

=

5

−1

Slope of BC =

⎣

⎢

⎢

⎡

5

−1

1

⎦

⎥

⎥

⎤

=5

∴ Slope of BC× Slope of altitude coming from A=−1

Equation of line through BC is,

y=5x+c

If B(−2,3) lies on above the line

i.e,3=−5×+C ⇒C=13

i.e,y=5x+13→ (1)

Similarly, the slope of AC =

3

2

Substituting (5,−1) in the equation

−1=

3

2

×5+C⇒C=−1

3

−10

=

3

−13

i.e,y=

3

2

x−

3

13

⇒3y=2x−13→(2)

eq Subtract Eqn (1) and (2)

0=[5x−

3

2

x]+[13−(

3

−13

)]

=

3

13x

+[

3

39+13

]=

3

13x+52

=0

⟹13x=−52i.e,x=−4

Put the value of x in eqn (1)

y=5(−4)+13=−7

∴ The third vertex is (−4,−7)