THEOREM 46.

The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.

Q

Let PT be a tangent at the point P to a circle whose centre is O.

It is required to prove that PT is perpendicular to the radius OP.

Proof.

Take any point Q in PT, and join OQ.

Then since PT is a tangent, every point in it except P is outside the circle.

.. OQ is greater than the radius OP.

And this is true for every point Q in PT;

.. OP is the shortest distance from O to PT.

Hence OP is perp. to PT. Theor. 12, Cor. 1.

Q.E.D.

COROLLARY 1. Since there can be only one perpendicular to OP at the point P, it follows that one and only one tangent can be drawn to a circle at a given point on the circumference.

COROLLARY 2. Since there can be only one perpendicular to PT at the point P, it follows that the perpendicular to a tangent at its point of contact passes through the centre.

COROLLARY 3. Since there can be only one perpendicular from O to the line PT, it follows that the radius drawn perpendicular to the tangent passes through the point of contact.

THEOREM 46. [By the Method of Limits.]

The tangent at any point of a circle is perpendicular to the radius drawn to the point of contact.

Let P be a point en a circle whose centre is O.

It is required to prove that the tangent at P is perpendicular to the radius OP.

Let RQPT (Fig. 1) be a secant cutting the circle at Q and P. Join OQ, OP.

Proof.

Because OP=OQ,

... the OQP = the OPQ;

.. the supplements of these angles are equal;

that is, the

OQR = the OPT,

and this is true however near Q is to P.

Now let the secant QP be turned about the point P so that Q continually approaches and finally coincides with P; then in the ultimate position,

(i) the secant RT becomes the tangent at P, Fig. 2, (ii) OQ coincides with OP;

and therefore the equal "OQR, OPT become adjacent,

.. OP is perp. to RT.

Q.E.D.

NOTE. The method of proof employed here is known as the Method of Limits.

THEOREM 47.

Two tangents can be drawn to a circle from an external point

Let PQR be a circle whose centre is O, and let T be an external point.

It is required to prove that there can be two tangents drawn to the circle from T.

Join OT, and let TSO be the circle on OT as diameter.

This circle will cut the OPQR in two points, since T is without, and O is within, the OPQR. Let P and Q he these points

Join TP, TQ; OP, OQ.

Proof. Now each of the TPO, TQO, being in a semicircle, is a rt. angle;

.. TP, TQ are perp. to the radii OP, OQ respectively.

›'. TP, TQ are tangents at P and Q.

Theor. 46.

Q.E.D,

COROLLARY. The two tangents to a circle from an external point are equal, and subtend equal angles at the centre.

1.

EXERCISES ON THE TANGENT.

(Numerical and Graphical.)

Draw two concentric circles with radii 50 cm. and 30 cm. Draw a series of chords of the former to touch the latter. Calculate and measure their lengths, and account for their being equal.

2. In a circle of radius 1'0′′ draw a number of chords each 16" in length. Shew that they all touch a concentric circle, and find its radius.

3. The diameters of two concentric circles are respectively 100 cm. and 50 cm. find to the nearest millimetre the length of any chord of the outer circle which touches the inner, and check your work by

measurement.

4. In the figure of Theorem 47, if OP=5", TO=13", find the length of the tangents from T. Draw the figure (scale 2 cm. to 5"), and measure to the nearest degree the angles subtended at O by the tangents.

X5. The tangents from T to a circle whose radius is 0.7" are each 2.4" in length. Find the distance of T from the centre of the circle. Draw the figure and check your result graphically.

(Theoretical.)

6. The centre of any circle which touches two intersecting straight lines must lie on the bisector of the angle between them.

7.

AB and AC are two tangents to a circle whose centre is O; shew that AO bisects the chord of contact BC at right angles.

8. If PQ is joined in the figure of Theorem 47, shew that the angle PTQ is double the angle OPQ.

9.

Two parallel tangents to a circle intercept on any third tangent a segment which subtends a right angle at the centre.

10. The diameter of a circle bisects all chords which are parallel to the tangent at either extremity.

11. Find the locus of the centres of all circles which touch a given straight line at a given point.

12. Find the locus of the centres of all circles which touch each of two parallel straight lines.

13. Find the locus of the centres of all circles which touch each of two intersecting straight lines of unlimited length.

14. In any quadrilateral circumscribed about a circle, the sum of one pair of opposite sides is equal to the sum of the other pair.

State and prove the converse theorem.

15. If a quadrilateral is described about a circle, the angles subtended at the centre by any two opposite sides are supplementary.

THEOREM 48.

If two circles touch one another, the centres and the point of contact are in one straight line.

TI

P

Let two circles whose centres are O and Q touch at the point P.

It is required to prove that O, P, and Q are in one straight line. Join OP, QP.

Proof. Since the given circles touch at P, they have a common tangent at that point.

Suppose PT to touch both circles at P.

Page 173.

Then since OP and QP are radii drawn to the point of

contact,

.. OP and QP are both perp. to PT ;

.. OP and QP are in one st. line. That is, the points O, P, and Q are in one st. line.

Theor. 2.

Q.E.D.

COROLLARIES. (i) If two circles touch externally the distance between their centres is equal to the sum of their radii.

(ii) If two circles touch internally the distance between their centres is equal to the difference of their radii.