Question

Prove that there is one and only one circle passing through three given non-collinear points...

please help me.....​

Answer 1

Answer:

Prove that there is one and only one circle passing trough three given non-collinear points.

Prove that there is one and only one circle passing trough three given non-collinear  points.

Theorem: There is one and only one circle passing through three given non-collinear points.

Given: Three non collinear points P, Q and R

To prove: There is one and only one circle passing through the points P, Q and R.

Construction: Join PQ and QR.

Draw perpendicular bisectors AB of PQ and CD of QR. Let the perpendicular bisectors intersect at the point O.

Now join OP, OQ and OR.

A circle is obtained passing through the points P, Q and R.

Proof: We know that, each and every point on the perpendicular bisector of a line segment is equidistant from its ends points.

Thus, OP = OQ  [Since, O lies on the perpendicular bisector of PQ]

and OQ = OR.  [Since, O lies on the perpendicular bisector of QR]

So,  OP = OQ = OR.

Let OP = OQ = OR =  r.

Now, draw a circle C(O,  r) with O as centre and  r  as radius.

Then, circle C(O,  r) passes through the points P, Q and R.

Next, we prove this circle is the only circle passing through the points P, Q and R.

If possible, suppose there is a another circle C(O′,  t) which passes through the points P, Q, R.

Then, O′ will lie on the perpendicular bisectors AB and CD.

But O was the intersection point of the perpendicular bisectors AB and CD.

So,  O ′ must coincide with the point O.  [Since, two lines can not intersect at more than one point]

As, O′P =  t  and OP =  r; and O ′ coincides with O, we get t = r .

Therefore, C(O,  r) and C(O,  t) are congruent.

Thus, there is one and only one circle passing through three the given non-collinear points.