Question

If the vertices of a triangle be (2, 1); (5,2) and (3,4) then its circumcentre is
​

Answer 1

Step-by-step explanation:

Given−

theco−ordinatesofthecircumcenteroftheΔPQR

withverticesP(x

1

,y

1

)=(2,1),Q(x

2

,y

2

)=(5,2)&R(x

3

,y

3

)=(3,4)

areC(x,y).

Tofindout−

C(x,y)=?

Solution−

HerePA=PB=PCsincetheyaretheradiiofthe

circumcentre.

∴PA

2

=PB

2

=PC

2

.Usingdistanceformula,

PA

2

=(x−2)

2

+(y−1)

2

,

PB

2

=(x−5)

2

+(y−2)

2

,

PC

2

=(x−3)

2

+(y−4)

2

.

PA

2

=PB

2

.

∴(x−2)

2

+(y−1)

2

=(x−5)

2

+(y−2)

2

⟹3x+y−12=0

⟹y=12−3x ...........(i).

AgainPB

2

=PC

2

.

∴(x−5)

2

+(y−2)

2

=(x−3)

2

+(y−4)

2

⟹−x+y+1=0

⟹y=x−1 ........(ii).

Compareing(i)&(ii),

12−3x=x−1

⟹x=

4

13

.

Puttingx=

4

13

in(ii),

y=

4

13

−1=

4

9

.

∴ThecircumcentreC(x,y)=(

4

13

,

4

9

).