Find the center of a circle that circumscribed about the triangle with vertices (0,-2),(7,-3) and (8,-2) .

Question

Find the center of a circle that circubscribed about the triangle with vertices (0,-2),(7,-3) and (8,-2) .

Answer 1

Let the circle is passing through the vertices of triangle P(0, - 2), Q(7, - 3), and R(8, - 2).

Let the center and radius of the circle be C(h, k) and r.

|PC| = |QC| = |RC|         (Since, the distance from the center and given three points are equal(radius))

|PC|² = |QC|² = |RC|²

(h - 0)² + (k + 2)² = (h - 7)² + (k + 3)² = (h - 8)² + (k + 2)²

(h - 0)² + (k + 2)² = (h - 7)² + (k + 3)²

h² + k² + 4k + 4 = h² - 14h + 49 + k² + 6k + 9

4k + 14h + 4 - 6k - 58 = 0

- 2k + 14h - 54 = 0

7h - k = 27

7h - k = 27 ----------> (1)

(h - 0)² + (k + 2)² = (h - 8)² + (k + 2)²

h² + k² + 4k + 4 = h² - 16h + 64 + k² + 4k + 4

4k + 4 = - 16h + 4k + 68

4 = - 16h + 68

16h = 68 - 4

16h = 64

→ h = 64/16 = 4.

Substitute h = 4 in equation (1).

7(4) - k = 27

28 - k = 27

→ k = 1.

Therefore, the center of the circle C(h, k) passing through the given points is (4, 1).