Page 4 - 10-Math-10 TANGENT TO A CIRCLE

110.. QTuaandgreanttictoEAquCaitricolnes eeLleeaarrnn..Ppuunnjjaabb

Theorem 2

10.1 (ii) The tangent to a circle and the radial segment joining
the point of contact and the centre are perpendicular to each
other.

Given: In a circle↔with centre O has
radius OC, Also AB is the tangent
to the circle at point C.

↔
To prove: AB and radial segment OC are perpendicular to each

other.

C↔onstruction: Take any point P other than C on the tangent line
AB . Join O with P so that OP meets the circle at D.

Proof:

Statements Reasons

↔ Given
AB is the tangent to the circle at point Construction
C. Whereas OP cuts the circle at D.

∴ mOC = mOP (i) Radii of the same circle

But mOD < mOP Point P is outside the
(ii) circle.

∴ mOC < mOP Using (i) and (ii)

So radius OC is shortest of all lines that

c↔an be drawn from O to the tangent line
AB

↔
Also OC AB

Hence, radial segment OC is ↔
perpendicular to the tangent AB .

Corollary: There can only be one perpendicular drawn to the radial
segment OC at the point C of the circle. It follows that one and only
one tangent can be drawn to the circle at the given point C on its
circumference.

Version 1.1 4