Find the coordinates of the circumcenter of △PQR with vertices P(−2,5), Q(4,1), and R(−2,−3).

Answer 1

midpoint of PQ is (1,3); slope of PQ is 4/-6 = -2/3, which means the perpendicular bisector has slope 3/2

so the perpendicular bisector of PQ has intercept B = y - mx = 3 - (3/2)(1) = 3 - 3/2 = 3/2

the perpendicular bisector of PQ is then y = (3/2)x + 3/2

midpoint of QR is (1,-1); slope of QR is 4/6 = 2/3, which means the perpendicular bisector has slope -3/2

so the perpendicular bisector of QR has intercept B = y - mx = -1 - (-3/2)(1) = -1 + 3/2 = 1/2

the perpendicular bisector of QR is then y = (-3/2)x + 1/2

midpoint of PR is (-2,1); slope of PR is undefined.

so the perpendicular bisector of PR is y = 1

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

(6/2)x + 3/2 = 1/2

3x + 3/2 = 1/2

3x = -1

x = -1/3

which forces y = (3/2)(-1/3) + 3/2 = -1/2 + 3/2 = 1

y = (-3/2)(-1/3) + 1/2 = 1

the circumcenter is (-1/3,1)

Answer 2

The circumcenter of a triangle is the point of intersection of the perpendicular bisectors. So we start by finding the midpoint and slope of each of the segments.

Midpoint of PQ = (1,3) Slope of PQ = -2/3

Midpoint of QR = (1, -1) Slope of QR = 2/3

Midpoint of PR = (-2, 1) Slope of PR = undefined

Use point-slope formula to find the equation of each line passing through the midpoint perpendicular to the line segment. Use the opposite reciprocal of each slope.

⊥PQ: y = (3/2)x+(3/2)

⊥QR: y = (-3/2)x + (1/2)

⊥PR: y = 1

Find point of intersection for 1st and 3rd equation:

Plug 1 for y in the first equation.

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

Subtract 3/2 to both sides.

-1/2 = 3/2 x

Divide both sides by 3/2.

-1/3 = x.

(-1/3, 1)

Find the point of intersection for the second and third equations.

Plug 1 for y in the 2nd equation.

1 = -3/2 x + 1/2

1/2 = -3/2 x

-1/3 = x

(-1/3, 1)

Since point of intersection is the same for both of the pairs, it follows by the transitive property that the first and second equations will share the same point of intersection. Thus, all 3 perpendicular bisectors intersect at a single point. Answer is (-1/3, 1)