Question

Orthocentre and centroid of a triangle are A(−3, 5) and B(3, 3) respectively. If C is the circumcentre and AC is the diameter of this circle, then find the radius of the circle

Answer 1

Let PQR be any triangle orthocentre, centroid and circumcentre.

An orthocentre is (−3, 5)

B centroid is (3, 3)

C orthocentre is (a, 6)

Also `"AB"/"BC" = 2/1`

B divides AC in the ratio 2 : 1

A line divides internally in the ratio point P is `(("m"x_2 + "n"x_1)/("m" + "n"), ("m"y_2 + "n"y_1)/("m" + "n"))`

m = 2

x₁ = 3

y₁ = 5

and

n = 1

x₂ = a

y₂ = b

∴ The point B `((2"a" - 3)/(2 + 1), (2"b" + 5)/(2 + 1))`

(3, 3) = `((2"a" - 3)/3, (2"b" + 5)/3)`

`(2"a" - 3)/3` = 3

2a – 3 = 9

2a = 9 + 3

2a = 12

a = `12/2` = 6

and 

`(2"b" + 5)/3`

2b + 5 = 9

2b = 9 – 5

2b = 4

b = `4/2` = 2

∴ Orthocentre C is (6, 2)

Diameter AC = `sqrt((6 + 3)^2 + (2 - 5)^2`

= `sqrt(9^2 + (-3)^2`

= `sqrt(81 + 9)`

= `sqrt(90)`

= `3sqrt(10)`

Radius = `(3sqrt(10))/2`

Radius = `3/2sqrt(10)`

or 

`3 xx sqrt(10/4) = 3sqrt(5/2)` units