Question

5. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre​

Answer 1

Answer:Let PT be a tangent to a circle with centre O, where P is the point of contact .

Let PQ perpendicular PT , where Q lies on the circle that is,

Angle QPT = 90°

If possible , let PQ not pass through the centre O

Join PO and produce it to meet the circle at R.

Then PO being the radius through the point of contact , we have

PO perpendicular PT

Angle OPT = 90°

Angle RPT = 90°

Thus we have Angle QPT =Angle RPT =90°

This is possible only if P ,Q and R are collinear .

But a straight line cuts a circle in at most two points.

So the point Q and R coincide .

Hence PQ passed through the centre O that is the perpendicular at the point of contact to the tangent passes through the centre .

Step-by-step explanation: