Topic 13-Symmetric Properties of Tangents

8/6/2019 Topic 13-Symmetric Properties of Tangents

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TOPIC 13

Properties of Tangents ofCircles


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AA tangenttangentlineline thatthat

intersectsintersects the circle atthe circle atexactlyexactly ONEONE pointpoint..

The wordThe word tangenttangent

comes from the Latincomes from the Latinword word tangenstangensmeaning meaning touchingtouching

y

A

B

CD

E


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Activity #1

1. In the given diagram,identify the chords andtangents.

2. The given diagram show acircle with centre O. Drawthe longest chord in thecircle. Write down thespecial name given to this

chord.

A

B

C

. O


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3. Use the diagram to complete the table.

T1 T3 T5

T2

T4

A

B

Tangent to the circle with Centre A

Tangent to the circle with centre B

Tangent to both circles


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4. The given diagram shows three circles that overlap one another. For eachpair of circles, name a line segment that is a chord to both circles.

P

Q

R

F

E

A

B

C

D


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Activity #2

Lets investigate the symmetryproperties of tangents to a circle.

1. Draw a circle and label itscentre as O.

2. Mark a point B on thecircumference of the circle.

3. Join the points O and B.

4. Draw the tangent CD to thecircle at B.5. Measure OBC and OBD.6. Also draw tangents FG and QR

at points E and P respectively.Measure the angles between

the radius and the tangent ineach case.

What is the relationship betweenthe tangent and the radius at the

point of contact?

.O

B

C

D

E

F

G

PQR

. .

.


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Activity #3

1. Draw a circle and label itscentre as O.

2. Mark a point C outside thecircle.

3. Draw the tangents CB and CDfrom point C to the circle.

4. Join OB, OD and OC.

5. Measure(a) the length of BC and DC,(b) OCB, OCD, COB andCOD

What conclusion can you draw?

.O

.C

D

B


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Tangent Properties

1. A tangent to a circleis perpendicular tothe radius through thepoint of contact

2. From an externalpoint, two tangentscan be drawn to a

circle, which are ofequal length.


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Example #1

TA and TB are tangents to the circle

with centre O. AOT = 50. Find thevalue ofxand y.

Solutionx = 90 (tangent B radius)

@

x = 90

ATO = 180 x 50 (sum of()= 40

y = ATO

= 40@y = 40

O

AB

x

y

T

50


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Example #2

TA and TB are tangents to the circle with

centre O, AT = 12 cm and OA = 5 cm. Findthe value ofx and y.

Solution

BT = AT (equal tangents from ext.pt.)@x = 12

In (AOT,A = 90 (tangent B radius)

y = 5 + 12 (Pythagorean Theorem)=169

@y = 169= 13

12 cmy cm

x cm

A

T

B

O


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Example #3In the given diagram, O is thecentre of the circle, AC is atangent to the circle at B and OAis parallel to BD. IfOAC = 38,find OBD.

Solution

Since OA / DB,DBC = OAC (corr. s)

= 38

OBC = 90 (tangent B radius)

Thus, OBD = 90 38 = 52

38

O

D C

B

A


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Example #4In the given diagram, O is the centre of the circle and

the line XY is a tangent to the circle at point X. If

XZW= WXY = 25, find(a)WXZ(b)XWZ(c)WYX

Solution(a) ZXY = 90 (tangent B radius)

WXZ = 90 - 25= 65

(b) XWZ = 180 - 25 - 65 ( sum of (WXZ)= 90

(c) WYX = 180 - 90 - 25 ( sum of (XYZ)= 65

Z

O

X

WY25

25


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Example #5

The diagram shows a circle with centre O. The lineXY is a tangent to the circle at point Z. IfZOW =

125r, find the values ofa. WYXb. OZV

Solution

a. YOZ = 180r 125 r ( sum of a straight line)= 55 r

OZY = 90 r (tangent B radius)

b. Since OZ and OV are radii of the circle, OZ = OV.Thus (OZV is an isosceles triangle.

OZV = (180 r - YOZ)= (180 r 55 r)= 62.5 r

125r

O

Y

X

ZW

V


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Example #6In the given diagram, O is the centre of the circle. AC andBC are tangents to the circle at A and B respectively. If

ACB = 80r

, finda. BAC b. AOB

Solutiona. Since AC and BC are tangents from the external point

C, AC = BC.Thus, (CAB is an isosceles triangle.

BAC = (180r - ACB)= (180 r - 80 r)= 50 r

b. Since CA and CB are tangents to the circle, OC bisectsACB.Thus, ACO = 80r z 2

= 40rCAO = CBO 90r (tangent B radius)CAO = 180r 90r - 40r

= 50rAOB = 2 x 50r

= 100r

A

B

O

C


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Activity #4

1. Given that AB and AC are tangents to the circle at Band C respectively and O is the centre of the circle, findthe value of x in each case.

48x

O B

C

A

110)O

B

A

C

(a)

(b)

O

B

A

C

55

x

O

B

A

C

80

) xx

(c)

(d)

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Topic 13-Symmetric Properties of Tangents