8.1 Tangents to a Circle

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8.1 Tangents to a Circle
 
1. A tangent to a circle is a straight line that touches the circle at only one point and the point is called the point of contact.


 
2. If a straight line cuts a circle at two distinct points, it is called a secant. The chord is part of the secant in a circle.



3. Tangent to a circle is perpendicular to the radius of the circle that passes through the point of contact.

If ABC is the tangent to the circle at B, then ∠ ABO = ∠ CBO = 90o. 
 
 

Properties of Two Tangents to a Circle from an External Point




In the diagram, BA and BC are two tangents from an external point B. The properties of the tangents are as follows.
 
(a) BA = BC
(b) ∠ ABO = ∠ CBO = xo
(c) ∠ AOB = ∠ COB = yo
(d) ∠ OAB = ∠ OCB = 90o
(e) ∠ AOC + ∠ ABC = 180o
(f) ∆ AOB and ∆ COB are congruent
 

SPM Practice (Short Questions)

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Question 1:


In figure above, FAD is a tangent to the circle with centre O. AEB and OECD are straight lines. The value of y is

Solution:
∠OAD = 90o
∠AOD= 180o  – 90o – 34o= 56o
y = 56o  ÷ 2 = 28o


Question 2:
In figure above, PQR is a tangent to the circle QSTU at Q and TUPV is a straight line. The value of y is

Solution:
∠QTS = ∠RQS = 40o
∠SQT= ∠QTS = 40o (isosceles triangle)
∠PQT= 180o  – 40o – 40o= 100o
∠TPQ= 180o  – 115o = 65o
y = 180o  – 100o – 65o= 15o


Question 3:


In figure above, ABC is a tangent to the circle BDE with centre O, at B.
Find the value of y.
 
Solution:
∠BOD= 2 × ∠BED
= 2 × 35o = 70o
∠ODB = ∠OBD
= (180o  – 70o) ÷ 2 = 55o

∠ EDB = ∠ EBA = 75o
yo + ∠ ODB = 75o
yo + 55o = 75o
y = 20o

8.2 Angle between Tangent and Chord

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8.2 Angle between Tangent and Chord

 
1. In the diagram above, ABC is a tangent to the circle at point B.
 
2. Chord PB divides the circle into two segments, that is, the minor segment PRB and the major segment PQB.

3. With respect to ∠PBA, ∠PQB is known as the angle subtended by chord BP in the alternate segment.

4. With respect to ∠QBC, ∠BPQ is known as the angle subtended by chord BQ in the alternate segment.
 
5. The angle formed by the tangent and the chord which passes through the point of contact of the tangent is the same as the angle in the alternate segment which is subtended by the chord.
 
6. The relationships between the angles are:
Angle ∠ABP = Angle ∠BQP
Angle ∠CBQ = Angle ∠BPQ

8.3 Common Tangents (Sample Questions)

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Example 1:

 
In the diagram, ABG and CDG are two common tangents to the circle with centres O and F respectively. Find the values of
(a) x,  (b) y,  (c) z.
 
Solution:
(a)
In ∆ BFG,
angle BFG = ½ × 56o = 28o
angle FBG = 90o ← (tangent is 90o to radius)
xo + 28o + 90o = 180o
xo = 180o – 118o
xo = 62o
x = 62
 
(b) 
angle AOF = xo = 62o ← (AO// BF)
yo = 2 × 62o
yo = 124o
y = 124

(c)
Exterior angle of AOC = 360o – 124o = 236o
Angle EOC = ½ × 236o = 118o
zo = (180o – 118o) × ½  ← (∆ EOC is isosceles triangle, angle OEC = angle OCE)
zo = 31o
z = 31

SPM Practice (Short Questions)

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Question 4:


In figure above, ABCD is a tangent to the circle CEF at point C. EGC is a straight line. The value of y is
 
Solution:
∠ C E F = ∠ D C F = 70 ∘ ∠ A E G + 70 ∘ + 210 ∘ = 360 ∘ ∠ A E G = 80 ∘ In cyclic quadrilateral A B G E , ∠ A B G + ∠ A E G = 180 ∘ y ∘ + 80 ∘ = 180 ∘ y = 100


Question 5:

In figure above, PAQ is a tangent to the circle at point A. AEC and BED are straight lines. The value of y is
 
Solution:
∠ABD = ∠ACD = 40o
∠ACB = ∠PAB = 60o
y= 180o  – ∠ACB – ∠CBD – ∠ABD
y= 180o  – 60o – 25o– 40o = 55o



Question 6:


In figure above, KPL is a tangent to the circle PQRS at point P. The value of x is

Solution:
∠PQS = ∠SPL= 55o
∠SPQ = 180o  – 30o – 55o= 95o
In cyclic quadrilateral,
∠SPQ + ∠SRQ = 180o
95o+ xo = 180o
x = 85o

8.3 Common Tangents (Part 1)

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8.3 Common Tangents
A common tangent to two circles is a straight line that touches each of the circles at only one point.
 
1. Intersect at two points
(a) Circles of the same size


Number of common tangents
Properties of common tangents
Two common tangents:
AB and CD
AC = BD
AB = CD
AB parallel to // OR parallel to // CD
 
 
(b) Circles of different sizes
 


Number of common tangents
Properties of common tangents
Two common tangents:
ABE and CDE
AB = CD
BE = DE
OA // RB
OC // RD

SPM Practice (Short Questions)

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Question 7:

In figure above, APB is a tangent to the circle PQR at point P. QRB is a straight line. The value of x is

Solution:
∠PQR = ∠RPB = 45o
∠QPR = (180o  – 45o) ÷ 2 = 67.5o
∠PQR + ∠BPQ + xo = 180o
45o + (67.5o + 45o) + xo = 180o
x = 22.5o



Question 8:

The figure above shows two circles with respective centres O and V. AB is a common tangent to the circles. OPRV is a straight line. The length, in cm, of PR is 

Solution:

cos 86 o = O M O V 0.070 = 1 O V O V = 1 0.070 O V = 14.29 c m ∴ P R = 14.29 − 5 − 4 = 5.29 c m

SPM Practice (Short Questions)

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Question 4:
John has a collection of stamps from Thailand, Indonesia and Singapore. He picks one stamp at random. The probability of picking a Thailand stamp is 1 5  and the probability of picking an Indonesia stamp is 8 15 .  John has 20 Singapore stamps. Calculate the total number of stamps in his collection. 

Solution:
Probability of picking a Singapore stamp = 1 − 1 5 − 8 15 = 4 15 Given that John has 20 Singapore stamps, ∴ Total number of stamp in John's collection = 15 4 × 20 = 75



Question 5:
A box contains 36 green erasers and a number of red erasers.
If an eraser is picked randomly from the box, the probability of picking a red eraser is 5 8 . Find the number of red erasers.

Solution:
Probability of picking a green eraser = 1 – 5 8 = 3 8

Total number of erasers in the box = 36 × 8 3 = 96

Hence, number of red erasers in the box = 96 – 36 = 60