10.1 Angle of Elevation and Angle of Depression (Part 2)

Posted on May 27, 2020 by Myhometuition

3. The angle of elevation or angle of depression is always measured from the horizontal line.

Example 3:

Diagram below shows two vertical poles JK and NL on a horizontal plane. M is a point on NL such that JK = ML.

The angle of depression of point J from point N is

Solution:

The angle of depression of point J from N is the angle between line JN and the horizontal line through N.

Angle of depression of J from N

= angle of elevation of N from J
=∠NJM

SPM Practice (Short Questions)

Posted on May 27, 2020 by Myhometuition

8.4 Circles, SPM Practice (Short Questions)

Question 9:

In figure above, ABC is a tangent to the circle centre O at B. AED is a straight line.

Find the value of y.

Solution:

∠ABO = 90^o

∠BOE = 2 × 40^o= 80^o

In quadrilateral AEOB,

∠AEO = 360^o– ∠ABO– ∠BOE – 35^o

= 360^o– 90^o – 80^o – 35^o= 155^o

y^o+ ∠AEO = 180^o

y^o+ 155^o = 180^o

y^o= 180^o– 155^o

y^o = 25

Question 10:

In figure above, ABC is a tangent to the circle centre O, at point B.

The value of x is

Solution:

∠OBC = 90^o

∠BOD = 2 × 50^o= 100^o

In quadrilateral BODC,

x^o= 360^o– ∠BOD – ∠OBC – 120^o

= 360^o– 100^o – 90^o – 120^o

= 50

8.3 Common Tangents (Part 3)

Posted on May 27, 2020 by Myhometuition

3. Do not intersect

(a) Circles of the same size

Number of common tangents	Properties of common tangents
Four common tangents: AB, CD, PS and RQ	AB = CD = OV PS = RQ AB // OV // CD

(b) Circles of different sizes

Number of common tangents	Properties of common tangents
Four common tangents: AB, CD, PS and RQ	AB = CD BT = DT PS = RQ OA // VB OC // VD

8.3 Common Tangents (Part 2)

Posted on May 27, 2020 by Myhometuition

2. Intersect at one point

(a) Circles of the same size

Number of common tangents	Properties of common tangents
Three common tangents: AB, CD and PQ	AC = PQ = BD AB = OR = CD AB // OR // CD AC // PQ // BD PQ perpendicular to OR

(b) Circles of different sizes

(i) External contact

Number of common tangents	Properties of common tangents
Three common tangents: ABE, CDE and PQ	AB = CD BE = DE OA // RB OC // RD PQ perpendicular to ORE

(ii) Internal contact

Number of common tangents	Properties of common tangents
One common tangent: PQR	ONQ perpendicular to PR

8.2 Angle between Tangent and Chord (Example 3 & 4)

Posted on May 27, 2020 by Myhometuition

Example 3:

In the diagram, PQR is a tangent to the circle QSTU at Q.

Find the values of

(a) x (b) y

Solution:

(a)

∠UTS + ∠UQS = 180^o ←(opposite angle in cyclic quadrilateral QSTU)

105^o + ∠ UQS = 180^o

∠ UQS = 75^o

x+ 75^o + 20^o = 180^o←(the sum of angles on a straight line PQR = 180^o)

x+ 95^o = 180^o

x = 85^o

(b)

∠ PQU = ∠ QSU ← (angle in alternate segment)

85^o = 35^o + y

y = 50^o

Example 4:

In the diagram, ABC is a tangent to the circle BDE with centre O, at B.

Find the value of x.

Solution:

\begin{array}{l} ∠ B E D = ∠ C B D = 54^{\circ} \\ ∠ B D E = \frac{180^{\circ} - 54^{\circ}}{2} = 63^{\circ} \leftarrow \begin{array}{l} Isosceles triangle \\ ∠ E B D = ∠ E D B \end{array} \\ ∠ A B E = ∠ B D E = 63^{\circ} \\ In △ A B E, \\ x^{\circ} + 45^{\circ} + 63^{\circ} = 180^{\circ} \\ x^{\circ} + 108^{\circ} = 180^{\circ} \\ x = 72 \end{array}

8.2 Angle between Tangent and Chord (Example 1 & 2)

Posted on May 27, 2020 by Myhometuition

Example 1:

In the diagram, ABC is a tangent to the circle BDE at B.

The length of arc BD is equal to the length of arc DE.

Find the value of p.

Solution:

Angle BED = 82^o ← (angle in alternate segment)

Angle DBE = 82^o ← (Arc BD = Arc DE, BDE is an isosceles triangle)

Therefore p= 180^o – 82^o – 82^o= 16^o

Example 2:

In the diagram, PQR is a tangent to the circle QSTU at Q.

Find the value of y.

Solution:

Angle QUT

= 180^o– 98^o ← (opposite angle in cyclic quadrilateral QSTU )

= 82^o

Angle QTU = 75^o ← (angle in alternate segment)

Therefore y= 180^o – (82^o + 75^o) ← (Sum of interior angles in ∆ QTU)

= 23^o

8.1 Tangents to a Circle (Examples)

Posted on May 27, 2020 by Myhometuition

Example 1:

In the diagram, O is the centre of a circle. ABC and CDE are two tangents to the circle at points B and D respectively. Find the length of OC.

Solution:

OC²= OB² + BC² ← (Pythagoras’ Theorem)

= 6² + 8²

= 100

OC = √100 = 10 cm

Example 2:

In the diagram, AB and BC are two tangents to a circle with centre O. Calculate the values of

(a) x (b) y

Solution:

(a)

AB = BC

7 + x = 12

x = 5

(b)

∠

OBA =

∠

OBC = 21^o

∠

OAB = 90^o ← (OA is perpendicular to AB)

y^o= 180^o – 21^o– 90^o

y = 69

Example 3:

In the diagram, ABC is a tangent to the circle with centre Oat point B. CDE is a straight line. Find the value of x.

Solution:

∠

CBO = 90^o ← (OB is perpendicular to BC)

In ∆ BCE,

x = 180^o – 30^o– 50^o– 90^o

x = 10^o

Question 6:
A box contains blue marbles and green marbles. A marble is picked at random from the box.
The probability of picking a blue marble is

\frac{6}{11}

. If there are 30 green marbles in the box, how many blue marbles are there in the box?

Solution:
Probability of picking a green marble = 1 –

\frac{6}{11}

\frac{5}{11}

Total number of marbles in the box = 30 ×

\frac{11}{5}

= 66

Hence, number of blue marbles in the box = 66 – 30 = 36

Question 7:

Table below shows the distribution of a group of 90 pupils attending a penalty kick training programme.

	Form Four	Form Five
Girls	33	15
Boys	18	24

A pupil is chosen at random from the group to start the penalty kick training.

What is the probability that a boy from Form Five will be chosen?

Solution:

P (a boy from Form Five)

= \frac{24}{90} = \frac{4}{15}

Question 8:

Diagram below shows some number cards.

A card is picked at random. State the probability that a prime number is picked.

Solution:

prime number = {11, 19, 37}
P (a prime number)

= \frac{3}{6} = \frac{1}{2}

SPM Maths

10.1 Angle of Elevation and Angle of Depression (Part 2)

SPM Practice (Short Questions)

8.3 Common Tangents (Part 3)

8.3 Common Tangents (Part 2)

8.2 Angle between Tangent and Chord (Example 3 & 4)

8.2 Angle between Tangent and Chord (Example 1 & 2)

8.1 Tangents to a Circle (Examples)

SPM Practice (Short Questions)

6.7 SPM Practice (Long Questions)

6.7 SPM Practice (Long Questions)