Mathematics Form 3

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MATHEMATICS FORM 3

- Amal Sufiah Akmal Shukri

- T

unku Kurshia

h Coll

ege

- Form 2N (2014)

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C O N T E N T S . . . C O N T E N T S . . .

•

• Chapter 1 - Lines and Angles IIChapter 1 - Lines and Angles II •

• Chapter 2 - Chapter 2 - PolPolygons IIygons II •

• Chapter 3 - Circles IIChapter 3 - Circles II •

• Chapter 4 - Statistics IIChapter 4 - Statistics II •

• Chapter 5 - IndicesChapter 5 - Indices •

• Chapter 6 - Algebraic Expressions IIIChapter 6 - Algebraic Expressions III •

• Chapter 7 - Algebraic FormulaeChapter 7 - Algebraic Formulae •

• Chapter 8 - Solid Geometry IIIChapter 8 - Solid Geometry III •

• Chapter 9 - Scale DrawingsChapter 9 - Scale Drawings •

• Chapter 10 Chapter 10 - Tr- Transformations IIansformations II •

• Chapter 11 - Linear Equations IIChapter 11 - Linear Equations II •

• Chapter 12 - Linear InequalitiesChapter 12 - Linear Inequalities •

• Chapter 13 - Graphs of FunctionsChapter 13 - Graphs of Functions •

• Chapter 14 - Chapter 14 - Ratios, Rates and Proportions IIRatios, Rates and Proportions II •

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C H A P T E R 1 C H A P T E R 1

L I N E S A N D A N G L E S I I L I N E S A N D A N G L E S I I

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1.1 ANGLES ASSOC

1.1 ANGLES ASSOCIAIATED WITH TED WITH PPARALLEL LINESARALLEL LINES

•

• Angles are the space within two lines or three orAngles are the space within two lines or three or

more planes diverging from a common point, or

within two planes diverging from a common line.

•

• Parallel lines are lines on the same plane thatParallel lines are lines on the same plane that

never meet, no matter how far they are extended.

•

• In this topic, we will learn In this topic, we will learn about transversal andabout transversal and

the angles formed can be

the angles formed can be classified asclassified as

corresponding angles, alternate angles or interior

angles

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Transversal

•

• A transversal is aA transversal is a

straight line that

intersects two or more

straight lines.

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• The figure on the rightThe figure on the right

shows two parallel lines

AC and DF intersected

by the transversal MN.

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Corr

Corresponding Angle

esponding Angle

s

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• CorresCorresponding angles ponding angles areare equal in size.

equal in size. •

• The figures on the rightThe figures on the right are the examples of

are the examples of corresponding angles. corresponding angles. •

• To identify correspondingTo identify corresponding angles, look for angles

angles, look for angles formed by lines that formed by lines that resemble the letter 'F'. resemble the letter 'F'.

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•

• Alternate angles are equalAlternate angles are equal in size.

in size.

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• The figures on the rightThe figures on the right shows the examples of

shows the examples of

alternate angles.

•

• To identify alternateTo identify alternate angles, look for angles

angles, look for angles

formed by lines that

resemble the letter 'Z'.

Alternate Angles

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• Interior angles are anglesInterior angles are angles

between parallel lines on the between parallel lines on the same side of

same side of transversal.transversal.

•

• The sum of two interior anglesThe sum of two interior angles

is 180 degrees. is 180 degrees.

•

• The figures on the right are theThe figures on the right are the

examples of interior angles. examples of interior angles.

•

• To identify interior angles, lookTo identify interior angles, look

for angles formed by lines that for angles formed by lines that resemble the letter 'C'.

resemble the letter 'C'.

Interior Angles

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EXAMPLE 1 EXAMPLE 1 C O R R E S P O N D I N G C O R R E S P O N D I N G A N G L E S A N G L E S

Find the value of x

Find the value of x in the diagramin the diagram above. above. Solution: Solution: 2x = 130 degrees 2x = 130 degrees Hence, Hence,

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EXAMPLE 2 EXAMPLE 2

AL

ALTERNTERNAATE ATE ANGLENGLESS

In the following diagram, BCD and ACE In the following diagram, BCD and ACE are straight lines. Fi

are straight lines. Find the value of nd the value of x.x.

Solution: Solution:

<AED = <BAE = 63

<AED = <BAE = 63 degreesdegrees

<BCE is the exterior angle for triangle <BCE is the exterior angle for triangle CDE.

CDE. Hence, Hence,

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EXAMPLE 3 EXAMPLE 3

I N T E R I O R A N G L E S I N T E R I O R A N G L E S

Find the value of x

Find the value of x in the diagramin the diagram above.

above. Solution: Solution:

<PQR + 117 degrees = 180

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C H A P T E R 2 C H A P T E R 2 POL

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•

• A polygon is a closed figA polygon is a closed figure with at least threeure with at least three

straight lines as its

straight lines as its sides.sides.

•

• An irregular polygon is one with unequal sides orAn irregular polygon is one with unequal sides or

interior angles.

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• The axis of symmetry of a polygon is a line thatThe axis of symmetry of a polygon is a line that

divides the polygon into

divides the polygon into two identical parts.two identical parts.

Polygons

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•

• A regular polygon is a polygon in which all sidesA regular polygon is a polygon in which all sides

are of equal length and all interior angles are of

equal size.

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• The number of axes of symmetry of a regularThe number of axes of symmetry of a regular

polygon is always the same as the number of its

sides.

2.1 REGULAR POLYGONS 2.1 REGULAR POLYGONS

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EXAMPLES OF

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Given that QPR is a straight Given that QPR is a straight line, then

line, then

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• u degrees is an interior angleu degrees is an interior angle

formed between two formed between two

adjacent sides PQ and PN. adjacent sides PQ and PN.

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• v degrees is an exterior anglev degrees is an exterior angle

formed between the adjacent formed between the adjacent side PN and the extended

side PN and the extended side PR of

side PR of the polygon.the polygon.

2.2 ANGLES

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•

• The sum of the The sum of the interior angle and the exterior angle interior angle and the exterior angle is 180 degrees.is 180 degrees. •

• The sum of the exterior angles of a The sum of the exterior angles of a regular polygon withregular polygon with nn sides is sides is

360 degrees.

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• The sum of the interior angles of The sum of the interior angles of a regular polygon witha regular polygon with nn sides issides is

(

(n - 2) x 180 degrees.n - 2) x 180 degrees.

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• In a regular polygon withIn a regular polygon with nn sides,sides,

a) the size of each interior angle

b) the size of each exterior angle

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EXAMPLE 1 EXAMPLE 1

In the

In the diagram below, AB, CDdiagram below, AB, CD, EF and GF , EF and GF areare straight lines.

straight lines.

Find the value of x. Find the value of x. Solution:

Solution:

The sum of exterior angle = 360 degrees The sum of exterior angle = 360 degrees

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EXAMPLE 2 EXAMPLE 2

The diagram shows an irregular polygon. The diagram shows an irregular polygon.

Find the value of x. Find the value of x. Solution:

Solution:

Given a polygon with 6 sides, so n = 6. Given a polygon with 6 sides, so n = 6. The sum of

The sum of interior anglesinterior angles

Hence, Hence,

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EXAMPLE 3 EXAMPLE 3

The diagram below shows a part of a regular polygon. The diagram below shows a part of a regular polygon.

Find the number of sides of the polygon. Find the number of sides of the polygon. Method 1

Method 1

Size of interior angle Size of interior angle 144 degrees = 144 degrees = then, then, Method 2 Method 2 Exterior angle Exterior angle Exterior angle Exterior angle then, then,

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C H A P T E R 3 C H A P T E R 3 C I R C L E S I I

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•

• A chord is a straight line connecting two points on the A chord is a straight line connecting two points on the circumference of acircumference of a

circle. circle.

•

• The diameter of a circle is The diameter of a circle is a chord that passes through the centre of a a chord that passes through the centre of a circle.circle. •

• The diameter of a circle divides the The diameter of a circle divides the circle into two equal parts.circle into two equal parts. •

• Any diameter of Any diameter of a circle is an axis of a circle is an axis of symmetry.symmetry.

•

• If the If the chord KL intersects perpendicularly with radius POchord KL intersects perpendicularly with radius PO, then KM = ML., then KM = ML.

3.1 SYMMETRY OF CIRCLES 3.1 SYMMETRY OF CIRCLES

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EXAMPLE EXAMPLE

The diagram shows a circle with centre The diagram shows a circle with centre O. The straight line UOV is

O. The straight line UOV is perpend

perpendicular to the choicular to the chord PUQ. rd PUQ. GivenGiven that PQ = 18cm and OV = 15cm.

that PQ = 18cm and OV = 15cm. Calculate the length of UV, in cm. Calculate the length of UV, in cm.

Solution: Solution:

Consider triangle OUQ, Consider triangle OUQ,

Hence, Hence,

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3.2 PROPERTIES OF ANGLES IN CIRCLES 3.2 PROPERTIES OF ANGLES IN CIRCLES

•

• The angle at the centre is The angle at the centre is twicetwice

the angle at the

the angle at the circumfercircumference.ence.

•

• The angles at theThe angles at the

circumfer

circumference subtended ence subtended byby the common arc are all equal. the common arc are all equal.

•

• The angle subtended at theThe angle subtended at the

circumfer

circumference in ence in a semicircle isa semicircle is a right angle.