The points A(1,3), B(7,1) & C(-3,-9) form a right angled triangle & lie on the circumference of a circle.What is the equation of the circle?

Slope of AB = ( 1 - 3)/(7 - 1)

= -2/6

= -1/3

Slope of AC = ( -9 - 3)/(-3 - 1)

= - 12/-4

= 3

Therefore AB and AC are perpendicular. Hence BC is a diameter of the circle. Diameter subtends 90 degs on the circumference

Centre of circle = midpoint of BC

= ( (7 - 3)/2, (- 9 + 1)/2)

= ( 2, - 4)

radius of circle r^2 = ( 2- 7)^2 + (-4 - 1)^2

= 25 + 25

= 50

Equation of circle

(x - 2)^2 + (y + 4)^2 = 50

Slope of AB = (3 - 1)/(1 - 7) = -1/3
Slope of AC = (3 + 9)/(1 + 3) = 3

(Slope of AB) * (Slope of AC) = -1
Then ∠BAC = 90°

Angle in semi-circle = 90°
Hence, BC is the diameter of the circle.

Center of the circle = ([7 - 3]/2, [1 - 9]/2) = (2, -4)
Diameter of the circle = BC = √[(7 + 3)² + (1 + 9)²] = 10√2
Radius of the circle = (10√2)/2 = 5√2

Equation of the circle :
(x - 2)² + (y + 4)² = (5√2)²
x² - 4x + 4 + y² + 8y + 16 = 50
x² + y² - 4x + 8y - 30 = 0

AB² = 6² + (-2)² = 40
BC² = (-10)² + (-10)² = 200
AC² = (-4)² + (-12)² = 160

BC² = AB² + AC²
BC is the diameter
Centre of BC is (2,-10)

[ x - 2 ]² + [ y + 10 ]² = r²

BC = 10 √2
r = 5 √2

[ x - 2 ]² + [ y + 10 ]² = 50

See Thales' theorem.
The hypoteneuse of such a triangle is a diameter of the circle.
AB^2 = 6^2 + 2^2 = 40
BC^2 = 10^2 + 10^2 = 200
AC^2 = 4^2 + 12^2 = 160
Thus the hypoteneuse is BC, the diameter is sqrt(200) = 10*sqrt(2), and the radius is 5*sqrt(2). The centre of the circle is the mid point of BC, ie 2, -4.
The equation of the circle is
(x - xcentre)^2 + (y - ycentre)^2 = radius^2
(x-2)^2 + (y+4)^2 = 50

The longest side is the hypotenuse, and that's the diameter of the circle.

AB = sqrt(6^2 + 2^2) = sqrt(40)
BC= sqrt(10^2 + 10^2) = sqrt(200)
AC = sqrt(4^2 + 12^2) = = sqrt(160)

So BC is longest, and you can see that AB^2 + AC^2 = BC^2 to verify that you do indeed have a right triangle.

The center of the circle is at the midpoint of the diameter: (B + C)/2 = (2, -4).

The square of the radius is 1/4 the square of the diameter: 200/4 = 50.

The equation of a circle centered at (a,b) with radius r is (x-a)^2 + (y-b)^2 = r^2, and with the above valuest, that makes your equation:

(x - 2)^2 + (y + 4)^2 = 50

Check to see that's true for each of the (x,y) pairs for A, B and C.

Distance AB

xAB = xB - xA = 7 - 1 = 6

yAB = yB - yA = 1 - 3 = - 2

AB² = xAB² + yAB² = (6)² + (- 2)² = 36 + 4 = 40 → AB = √40


Distance BC

xBC = xC - xB = - 3 - 7 = - 10

yBC = yC - yB = - 9 - 1 = - 10

BC² = xBC² + yBC² = (- 10)² + (- 10)² = 100 + 100 = 200 → BC = √200


Distance AC

xAC = xC - xA = - 3 - 1 = - 4

yAC = yC - yA = - 9 - 3 = - 12

AC² = xAC² + yAC² = (- 4)² + (- 12)² = 16 + 144 = 160 → AC = √160


You can see that [BC] is the biggest. As the triangle A, B and C is a right triangle, you can say that [BC] is the hypotenuse, so the midpoint of [BC] is the center of the circle. This is the point M.

xM = (xB + xC)/2 = (7 - 3)/2 = 2

yM = (yB + yC)/2 = (1 - 9)/2 = - 4

…and you can say that [BC] is the diameter of the circle.


BC = √200 = 10√2 ← this is the diameter, and to get the radius, you divide by 2

R = 5√2


The typical equation of a circle is: (x - xo)² + (y - yo)² = R² → where:

xo: abscissa of center → 2 in your case

yo: ordinate of center → - 4 in your case

R: radius of circle → 5√2 in your case → R² = 25 * 2 = 50


→ The equation of your circle is: (x - 2)² + (y + 4)² = 50

WE