Points P (4,1) , Q(6,5) , R(2,7) lie on a circle . What is the perpendicular distance of the chords PQ , QR and PR from the centre ??

Dear Student,

We form our diagram from given information , As :

Here , OA $\perp$ PQ  , OB $\perp$ QR and OC $\perp$ PR and we know perpendicular from center to any chord also bisects the chord , So

A , B and C are mid points of PQ , QR and PR respectively .

We know formula for mid point when coordinate of two end points given :

So,

Coordinate of A = ,

Coordinate of B =

And

Coordinate of C =

We know slope of equation when coordinate of two end points given :
So,

Slope of line PQ =

And we know product of slope of two perpendicular lines will give  = - 1  , If we assume slope of line OA = m2 , So

m1 m2 =  -  1  , Substitute value of ' m1 '  as we calculated above and get :

( 2 ) m2 = - 1 ,

m2  = $-\frac{1}{2}$

We know equation of line when we know slope of line and a point of coordinate where that passing through :

( y y1 ) = m ( x  - x1 )

Then for line OA :  x1 = 5 ,  y1 = 3 and m  = $-\frac{1}{2}$ , So

( y - 3 ) = $-\frac{1}{2}$ ( x  - 5 ) ,

2 y  - 6 = - + 5  ,

x + 2 y  =  11                                                                              --- ( 1 )

And Slope of line QR =

And we know product of slope of two perpendicular lines will give  = - 1  , If we assume slope of line OB = m2 , So

m1 m2 =  -  1  , Substitute value of ' m1 '  as we calculated above and get :

( $-\frac{1}{2}$ ) m2 = - 1 ,

m2  = 2

Then for line OB :  x1 = 4 ,  y1 = 6 and m  = 2 , So

( y - 6 ) = 2 ( x  - 4 ) ,

y  - 6 = 2 - 8  ,

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

Now we multiply by 2 in equation 2 and get :

4x - 2 y  =  4                                                                             --- ( 3 )

Now we add equation 1 and 3 and get :

5 x  =  15  ,

x  =  3 , Substitute that value in equation 1 and get

3  + 2 y  =  11  ,

2 y  = 8   ,

y  = 4

So,

Coordinate of center  =  (  3 , 4 )

We know distance formula :  d  =

For perpendicular distance of the chords PQ from center , x1 = 3 ,  y1 =  4 and x2 = 5 ,  y2 = 3 , So

OA  =

And

For perpendicular distance of the chords QR from center , x1 = 3 ,  y1 =  4 and x2 = 4 ,  y2 = 6 , So

OB  =

And

For perpendicular distance of the chords PR from center , x1 = 3 ,  y1 =  4 and x2 = 3 ,  y2 = 4 , So

OC  =