Question

Find the circumcentre of the triangle formed by the lines 3x-y-5=0,x+3y-5=0,x=y

Answer 1

Heya User,

--> 3x - y = 5 --------> (i)
--> x + 3y = 5 -------> (ii)
--> x = y --------------> (iii)

From (i) and (ii),
---> A ≡  (2,1)
From (ii) and (iii),
---> B ≡ (1.25,1.25)
From (i) and (iii),
---> C ≡ (2.5,2.5)

Now, we find midpoints :- D,E,F of sides AB,BC,CA respectively:->
--> D ≡ (1.625, 1.125)
--> E ≡ (1.875, 1.875)
--> F ≡ (2.25, 1.75)

Now, Slopes of :->
:-> (i) = 3
:-> (ii) = -1/3
:-> (iii) = 1

=> Slopes of line perpendicular to :->
:-> (i) = -1/3
:-> (ii) = 3
:-> (iii) = -1

=> Equations of line :->
:-> DO -> ( y - 1.625 ) = -1/3 ( x - 1.125 )
=> DO -> 3y - 4.875 = 1.125 - x
=> DO -> 3y + x - 6 = 0 --> (iv)

Similarly, we find : FO -> ( y - 2.25 ) = ( 1.75 - x )
=> FO -> y + x - 4 = 0 --> (v)

Solving (iv) and (v),
--> O ( x , y ) ≡ ( 3 , 1 ) <--- The Circumcenter

Answer 2

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