8 8.1Tangents to a Circle 8.2Tangents to a Circle from an External Point Chapter Summary Case Study 8.3Angles in the Alternate Segments Basic Properties

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8.1 Tangents to a Circle

8.2 Tangents to a Circle from an External Point

Chapter Summary

Case Study

8.3 Angles in the Alternate Segments

Basic Properties of Circles (2)

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Case Study

The wheels of a train and the rails illustrate an important geometrical relationship between circles and straight lines.

When the train travels on the rails, it shows how a circle and a straight line touch each other at only one point.

Can you give me onereal-life example of a circle and a straight line?

Yes, the wheel of a train is a circle and the rail is a straight line.

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8.1 Tangents to a Circle8.1 Tangents to a Circle

We can draw a straight line AB and a circle in three different ways:Case 1: The straight line does not meet the circle.

Case 2: The straight line cuts the circle at two distinct points, P and Q.

Case 3: The straight line touches the circle at exactly one point, T.

In case 3, at each point on a circle, we can draw exactly one straight line such that the line touches the circle at exactly one point.

Tangent to a circle:straight line if and only if touching the circle at exactly one point

Point of contact (point of tangency):point common to both the circle and the straight line

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There is a close relationship between the tangent to a circle and the radius joining the point of contact:

Theorem 8.1If AB is a tangent to the circle with centre O atT, then AB is perpendicular to the radius OT.Symbolically, AB OT. (Reference: tangent radius)

This theorem can be proved by contradiction:

Suppose AB is not perpendicular to the radius OT.Then we can find another point T on AB such that OT AB.Using Pythagoras’ Theorem, OT is shorter than OT. Thus T lies inside the circle.

∴ AB cuts the circle at more than one point.

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The converse of Theorem 8.1 is also true:

Theorem 8.2OT is a radius of the circle with centre O andAB is a straight line that intersects the circleat T. If AB is perpendicular to OT, then ABis a tangent to the circle at T.In other words, if AB OT,

then AB is a tangent to the circle at T. (Reference: converse of tangent radius)

The perpendicular to a tangent at its pointof contact passes through the centre ofthe circle.

Hence we can deduce an important fact:

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Example 8.1T

Solution:

In the figure, O is the centre of the circle. AB is a tangent to the circle at T. OC TC 9 cm. (a) Find CAT and CTA. (b) Find the length of AT.

(a) OT OC 9 cm (radii)

∴ OCT is an equilateral triangle. ∴ COT OTC 60 (prop. of equilateral )

OTA 90 (tangent radius)

In OAT,CAT OTA COT 180 ( sum of )

∴ CTA 90 60 30

CAT 30

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Example 8.1T

Solution:

In the figure, O is the centre of the circle. AB is a tangent to the circle at T. OC TC 9 cm. (a) Find CAT and CTA. (b) Find the length of AT.

(b) ∵ CTA CAT 30 (proved in (a))

∴ CA CT 9 cm (sides opp. equal s)

In OAT,AT 2 OT 2 OA 2 (Pyth. Theorem)

AT cm22 918 cm39

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Example 8.2T

Solution:

In the figure, AB is a tangent to the circle at T. POQB is a straight line. If ATP 65, find TBQ.

OTA 90 (tangent radius)

∴ OTP 90 65 25

Join OT.

∵ OP OT (radii)

∴ OPT OTP 25 (base s, isos. )In BPT,ATP TBQ OPT (ext. of )

65 TBQ 25TBQ 40

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8.2 Tangents to a Circle from8.2 Tangents to a Circle from an External Point an External Point

Consider an external point T of a circle.We can always draw two tangents from that point.

In the figure, we can prove that OTA OTB (RHS):

OAT OBT 90 (tangent radius)

OT OT (common side)

OA OB (radii)

Hence the corresponding sides and the corresponding angles of OTA and OTB are equal:

TA TB (corr. sides, s)

TOA TOB (corr. s, s)

OTA OTB (corr. s, s)

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Properties of tangents from an external point:

Theorem 8.3In the figure, if TA and TB are the two tangentsdrawn to the circle with centre O froman external point T, then(a) the lengths of the two tangents are

equal, that is, TA TB; (b) the two tangents subtend equal angles at the centre,

that is, TOA TOB; (c) the line joining the external point to the centre of the

circle is the angle bisector of the angle included by the two tangents, that is, OTA OTB.

(Reference: tangent properties)

In the figure, OT is the axis of symmetry.

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Example 8.3T

Solution:

In the figure, TA and TB are tangents to the circle with centre O. If ABT 65, find(a) ATB, (b) AOB.

(a) ∵ TA TB (tangent properties)

∴ TAB TBA (base s, isos. )

65In TAB,ATB 2(65) 180 ( sum of )

ATB 50

(b) OAT OBT 90 (tangent radius)

∴ AOB OAT ATB OBT 360 ( sum of polygon)

AOB 90 50 90 360 AOB 130

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Example 8.4T

Solution:

In the figure, TA and TC are tangents to the circle with centre O.

If AB : BC 1 : 2 and ADC 66, find x and y.

( (

ABC 66 180 (opp. s, cyclic quad.)

ABC 114ACB : BAC AB : BC (arcs prop. to s at ⊙ce)

( (

x : BAC 1 : 2∴ BAC 2x

In ABC,ABC BAC x 180 ( sum of )

114 2x x 180 x 22

2x

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Example 8.4T

Solution:

In the figure, TA and TC are tangents to the circle with centre O.

If AB : BC 1 : 2 and ADC 66, find x and y.

( (

AOC 2 66 ( at the centre twice at ⊙ce)

OAT OCT 90 (tangent radius)

132

∴ AOC OAT ATC OCT 360 ( sum of polygon)

132 90 ATC 90 360 ATC 48

∵ TC TA (tangent properties)

∴ TCA TAC (base s, isos. )

In TAC,ATC 2TAC 180 ( sum of )

TAC 66

∵ BAC 2x 44

∴ y 22

2x

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Example 8.5T

Solution:

The figure shows an inscribed circle in a quadrilateral ABCD.If AB 16 cm and CD 12 cm, find the perimeter of the quadrilateral.

Referring to the figure,AP AS, BP BQ, CQ CRand DR DS. (tangent properties)

S

P

Q

R

Let AP AS a, BP BQ b, CQ CR c and DR DS d.Then a b 16 cm and c d 12 cm.∵ DA AS SD

a dand BC BQ QC

b c ∴ Perimeter 16 cm (b c) 12 cm (a d)

16 cm 12 cm a b c d 56 cm

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8.3 Angles in the Alternate Segments8.3 Angles in the Alternate Segments

In the figure, AB is a tangent to the circle at T and PT is a chord of the circle.

Tangent-chord angles:angles formed between a chord and a tangent to a circle, such as PTA and PTB.

Alternate segment:segment lying on the opposite side of a tangent-chord angle

segment I is the alternate segment with respect to PTB segment II is the alternate segment with respect to PTA

Consider the tangent-chord angle b.Then a is an angle in the alternate segment with respect to b.

Notes:We can construct infinity many angles in the alternate segment with respect to b.

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The figure shows another angle in the alternate segment with respect to b with BR passing through the centre O.

R C a ( in the same segment)RAB 90 ( in semicircle)

In ABR,R RAB ABR 180 ( sum of )

ABR 90 a∵ ABR ABQ 90 (tangent radius)∴ (90 a) b 90

a b

a 90 ABR 180

Theorem 8.4A tangent-chord angle of a circle is equalto any angle in the alternate segment.Symbolically, a b and p q.(Reference: in alt. segment)

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Example 8.6T

Solution:

In the figure, TS is a tangent to the circle. TBC is a straight line. BA BT and ATB 48. (a) Find ACB. (b) Find CAS.

(a) ∵ BA BT (given)

∴ BAT BTA (base s, isos. )

48 ∴ ACB BAT ( in alt. segment)

48

(b) CBA BTA BAT (ext. of )

96 ∴ CAS CBA ( in alt. segment)

96

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Example 8.7T

Solution:

The figure shows an inscribed circle of ABC. The circle touches the sides of the triangle at P, Q and R respectively. If BAC 40 and ACB 68, find all the angles in PQR.

∵ AP AR (tangent properties)

∴ APR ARP (base s, isos. )

In PAR,40 APR ARP 180 ( sum of )

ARP 70

Similarly, ∵ CQ CR (tangent properties)

∴ CRQ CQR 56

∴ PQR ARP 70 ( in alt. segment)

∴ QPR CRQ 56 ( in alt. segment)

∴ PRQ 180 70 56 54

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The converse of Theorem 8.4 is also true:

Theorem 8.5A straight line is drawn through an endpoint of a chord of a circle. If the anglebetween the straight line and the chordis equal to an angle in the alternatesegment, then the straight line is atangent to the circle. In other words, if x y, then TA is atangent to the circle at A. (Reference: converse of in alt. segment)

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Example 8.8T

Solution:

In the figure, AB // PQ and CD is a common chord of the circles. Prove that PQ is a tangent to the larger circle.

BAC CQP (alt. s, AB // PQ)

BAC CDQ (ext. , cyclic quad.)

∴ CQP CDQ

∴ PQ is a tangent to the larger circle. (converse of in alt. segment)

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8.1 Tangents to a Circle

Chapter Summary

1. If AB is a tangent to the circle with centre O at T, then AB is perpendicular to the radius OT.

(Ref: tangent radius)

2. OT is a radius of the circle with centre O and ATB is a straight line. If AB is perpendicular to OT, then AB is a tangent to the circle at T.

(Ref: converse of tangent radius)

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Chapter Summary8.2 Tangents to a Circle from an External Point

If TA and TB are tangents to the circle with centre O,from an external point T, then(a) TA TB;

(The length of the two tangents are equal.)(b) TOA TOB;

(Two tangents subtend equal angles at the centre.)(c) OTA OTB.

(OT bisects the angle between the two tangents.)

(Ref: tangent properties)

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Chapter Summary8.3 Angles in the Alternate Segments

1. If TA is a tangent to the circle, then x y and p q.

(Ref: in alt. segment)

2. If x y, then TA is a tangent.(Ref: converse of in alt. segment)

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8 8.1Tangents to a Circle 8.2Tangents to a Circle from an External Point Chapter Summary Case Study 8.3Angles in the Alternate Segments Basic Properties