Question

find the center for (1,4) (1,2) (6,2).

Answer 1

 Let the points are A(1,4), O(1,2), B(6,2)

Here we can solve the center use the graph.

Plot the points and connect it.

AOB is right angle traingle.

The three altitutdes of triangle meet in one point called the ortho center.

If it is right triangle the ortho center is vertex which is the right angle.

From the graph ortho center is (1,2)

The three perpendicular bisectors of the sides of a triangle meet in one point called the circum center.

Here the circumcenter is mid point of the hypotenuse.

(x,y) = (6,4)  and ortho center (1,2)

The circum centere is E = [(6+1)/2, (4+2)/2]

The circum centere is E = [7/2,6/2]

E = (3.5,3)

Answer 2

Let the circle is passing through the points P(1, 4), Q(1, 2), and R(6, 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 - 1)² + (k - 4)² = (h - 1)² + (k - 2)² = (h - 6)² + (k - 2)²

(h - 1)² + (k - 4)² = (h - 1)² + (k - 2)² → k = 3.

(h - 1)² + (k - 2)² = (h - 6)² + (k - 2)²  → h = 35/10 = 3.5.

Therefore, the center of the circle C(h, k) passing through the given points is (3.5, 3).