Maths Help?

Question

The three vertices of a triangle are; K (4, 1), L (-1, 6) and M (- 4, -3). If a circle is drawn with K, L and M on the circumference (ie. the circumcircle of the triangle), find the coordinates of the centre of the circle.

Answer 1

From the given data draw the below diagram :

Observe the diagram, KN and LO are the perpendecular bisectors to two of the sides of the triangle KLM.

The intersection point of KN and LO is the center of the circle.

First find the equations for perpendecular bisectors.

KN is  perpendecular bisector to LM.

• Slope of the line LM : (- 3 - 6)/(- 4 - (- 1)) = (- 9)/(- 3) = 3.

Since, KN is  perpendecular bisector to LM, slope of KN is - 1/3.

And mid point of LM = [(- 1 - 4)/2, (6 - 3)/2] = (- 5/2, 3/2).

The perpendecular bisector KN has the slope (m) = - 1/3 and passes through the mid point (- 5/2, 3/2).

The equation for KN : y - 3/2 = (- 1/3)(x + 5/2).

3(2y - 3) = - 2x - 5

6y - 9 = - 2x - 5

2x + 6y = 4

x + 3y = 2.

⇒ KN = x + 3y = 2 → ( 1 )

• Slope of the line KM : (- 3 - 1)/(- 4 -  4) = (- 4)/(- 8) = 1/2.

Since, LO is  perpendecular bisector to KM, slope of LOis - 2.

And mid point of KM = [(4 - 4)/2, (1 - 3)/2] = (0, - 1).

The perpendecular bisector LO has the slope (m) = - 2 and passes through the mid point (0, - 1).

The equation for LO : y + 1 = (- 2)(x - 0).

y + 1 = - 2x

⇒ LO = 2x + y = - 1 → ( 2 )

• Solve equation (1) and (2).

Simplify equation (2) : 2x + y = - 1

y = - 1 - 2x.

Substitute y = - 1 - 2x in equation (1) : x + 3y = 2.

x + 3(- 1 - 2x) = 2

x - 3 - 6x = 2

- 5x = 5

⇒ x = - 1.

Substitute x = - 1 in either of two equations.

Equation 1 : x + 3y = 2

- 1 + 3y = 2

3y = 3

⇒ y = 1.

Solution is (- 1, 1), this is the intersection point of perpendecular bisectors KN and LO.

The intersection point of KN and LO is the center of the circle.

Therefore, (- 1, 1) is the center of the circle.