year 12 further mathematics - paul...

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Year 12 Further Mathematics

UNIT 3

MODULE 2: Geometry and Trigonometry

Chapter 7

This module covers the application of geometric and trigonometric knowledge and techniques to various two-‐dimensional and three-‐dimensional practical spatial problems.

Familiarity with the trigonometric ratios sine, cosine and tangent, similarity and congruence, pythagoras theorem, basic properties of triangles and applications to regular polygons, corresponding, alternate and co-‐interior angles and angle properties of regular polygons is assumed.

Geometry, including:

• Pythagoras theorem in two and three dimensions and the use and applications of similarity;

• Calculation of surface area and volume of regular and composite solids;

• Application of the effect of changing linear dimensions (that is, if the linear scale factor is k,

then the area scale factor is k2 and the volume scale factor is k3).

Module 2: Geometry and Trigonometry

7A 1, 2, 3, 4, 5, 6, 7 7B 1, 3, 5, 7, 8, 9, 10, 11 7C 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12 7D 1, 2, 3(a, b), 4 (a , b, c), 5(a, b, c), 6(a, c ,e), 7(a, c, e), 9, 10, 11, 12, 13, 14 7E 1, 2, 3, 4, 6, 7, 8 7F 1, 2(a, c, e), 3(a, c, e), 4, 6, 7, 8, 9, 10 7G 1, 2, 3, 4, 5, 8, 9, 10, 12, 14, 15, 16, 17

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Table of Contents MODULE 2: Geometry and Trigonometry ....................................................................................... 1

Table of Contents ............................................................................................................................... 2

7A PROPERTIES OF ANGLES, TRIANGLES AND POLYGONS ........................................... 3

Interior Angles of Polygons ....................................................................................................... 3

Geometry rules, definitions and notation rules .......................................................................... 3

7B AREA AND PERIMETER ...................................................................................................... 6

Perimeter .................................................................................................................................... 6

Area ............................................................................................................................................ 7

Composite Areas ........................................................................................................................ 9

7C TOTAL SURFACE AREA .................................................................................................... 10

Total surface area formulas of common objects ...................................................................... 10

Total surface area using a net .................................................................................................. 11

7D VOLUME OF PRISMS, PYRAMIDS AND SPHERES ....................................................... 13

Unit conversion for volume ...................................................................................................... 13

Volume of prisms ...................................................................................................................... 13

Volume of Pyramids ................................................................................................................. 15

Volume of sphere ...................................................................................................................... 15

7E SIMILAR FIGURES .............................................................................................................. 17

Scale Factor, k ......................................................................................................................... 17

7F SIMILAR TRIANGLES ......................................................................................................... 19

Similar triangle properties ....................................................................................................... 19

7G AREA AND VOLUME SCALE FACTORS ........................................................................ 22

Area of similar figures ............................................................................................................. 22

Volume of similar figures ......................................................................................................... 23

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7A PROPERTIES OF ANGLES, TRIANGLES AND POLYGONS

Interior Angles of Polygons For a regular polygon (all sides and angles equal) of n sides,

• Interior angle n

oo 360180 −=

• Exterior angle n

o360=

Example 1: Find the interior and exterior angle of the regular polygon shown below.

Geometry rules, definitions and notation rules Definitions of common terms

Some common notations and rules

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Example 2: Find the values of the pronumerals in polygon at below.

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Example 3: Find the missing pronumerals in the diagram of railings for a set of stairs shown at below.

Identify/recognize that the diagram is made up of a series of parallel lines.

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7B AREA AND PERIMETER Unit Conversion for Length

Perimeter Perimeter is the distance around a closed figure. To find the perimeter for any shapes with straight edges, we simply add up all the edges provided that they are all written in the same unit. Perimeter of common shapes:

Example 1: Calculate the perimeter for the following: 1. P = 2. C =

6 m

400 cm

14 cm

× 1000 × 100 × 10

÷ 1000 ÷ 100 ÷ 10

km m cm mm

d r = radius d = diameter r = ½ d

= 2l + 2w

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3. P =

4. P =

Area Area is the space inside a two dimension shape. Areas are measured in square units. Unit Conversion for Area Area of Common Shapes

7 cm

7 cm

5 cm

4 cm

× 10002 × 1002 × 102

÷ 10002 ÷ 1002 ÷ 102

km2 m2 cm2 mm2

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Example 2: Calculate the area for the following: 1. 2. 3.

12 cm

7 cm

5 cm

3 cm

2 cm

3.5 cm

A =

A =

A =

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Composite Areas Figures that are made up of combination between common shapes are generally known as composite figures. The area for composite figures can be calculated by calculating for the area of each common shape and then adding these areas up. Area of a composite figure = sum of the areas of the individual common figures A(composite) = A1 + A2 + A3 + A4 + . . . Example 6: Find the area of the hotel foyer from the given below (to the nearest square metre).

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7C TOTAL SURFACE AREA The total surface area (TSA) of a solid object is the sum of the areas of the surfaces.

Total surface area formulas of common objects

Example 9: Find the total surface area of a poster tube with a length of 1.13 m and a radius of 5 cm. Give your answer to the nearest 100 cm2. Example 10: Find the total surface area of a size 7 basketball with a diameter of 25 cm. Give your answer to the nearest 10 cm2.

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Example 11: A die used in a board game has a total surface area of 1350 mm2. Find the linear dimensions of the die (to the nearest mm).

Total surface area using a net When ask to determine the total surface area of non common object, it is easier to construct a net of the object then calculate the sum of the all individual net area.

The total surface area of the above net is calculated by finding the area of the four triangles plus the area of the square base adding together. Example 12: Find the total surface area of the triangular prism shown below.

Constructing the net for the triangular prism

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Calculate TSA of the triangular prism Example 13: Find the surface area of an open cylindrical can that is 12 cm high and 8cm in diameter (to 1 decimal place).

The net of the opened cylinder

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7D VOLUME OF PRISMS, PYRAMIDS AND SPHERES

Unit conversion for volume Example: Convert the following to the unit indicated. (a) 1.12 cm3 (mm3) = _____________________________________________________

(b) 0.658 × 10−

5 m3 (mm3) = _______________________________________________

(c) 0.4735 L (mm3) = _____________________________________________________

(d) 0.156 m3 (L) = _______________________________________________________

Volume of prisms A prism is a three dimensional object that has a uniform cross-section. A prism is named accordance with its uniform cross-section area.

The volume of the prism can be calculated by the formula below: Volume of prism = Area of uniform cross-section × height = A × H

× 1003 × 103

÷ 1003 ÷ 103

m3 cm3 mm3

× 1000

÷ 1000

L mL 1 cm3 = 1 mL

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Example 16: Find the volume of the object (to the nearest cm3). V(cylinder) = Example 17: Find the volume of the slice of bread with a uniform cross-sectional area of 250 mm2 and a thickness of 17mm (to the nearest mm3). V = A(cross-section) × H Example 18: Find the height of the triangular prism from the information provided in the diagram at right (to 1 decimal place).

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Volume of Pyramids A pyramid is a three dimensional object that has a similar cross-section but the size reduces as it approaches the vertex.

The name of the pyramid is related to its similar cross-sectional area or base. Note a circular pyramid is called a cone.

Volume of pyramid = 31 × area of cross-section at the base × height

V = 31 × A(base cross-section) × H

The height of a pyramid, H, is sometimes called the altitude. Example 19: Find the volume o the pyramid below to the nearest m3.

V(pyramid) = 31 × A(square) × H

Volume of sphere

V(sphere) = 34πr3

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Example 20: Find the volume of the object shown below to the nearest litre.

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7E SIMILAR FIGURES Two objects that have the same shape but different size are said to be similar. When comparing shapes, the starting shape is referred as the original shape and the transformed shape as the image. Image is denoted by the symbol Aʹ′ . For two figures to be similar, they must have the following properties: 1. The ratios of the corresponding sides must be equal.

RatioCommon AD

DACD

DCBC

CBAB

BA=ʹ′ʹ′

=ʹ′ʹ′

=ʹ′ʹ′

=ʹ′ʹ′

2. The corresponding angles must be equal.

O

O

O

O

DDCCBBAA

906012585

=ʹ′∠=∠

=ʹ′∠=∠

=ʹ′∠=∠

=ʹ′∠=∠

Scale Factor, k 1. Scale factor, k, is the amount of enlargement or reduction and is expressed as integers, fraction

or scale ratios.

For example, k = 2, k = ½ , k = 1: 2

2. CA

ACBC

CBAB

BAOriginal ofLength Image ofLength kfactor, Scale

ʹ′ʹ′=ʹ′ʹ′

=ʹ′ʹ′

==

• For enlargement, k is greater than one (k>1). • For reduction, k is between zero and one (0<k<1).

3. For k = 1, the figures are exactly the same shape and size and are referred to as congruent.

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Example 21: For the similar shapes shown on the right:

(a) Find the scale factor for the reduction of the shape (b) Find the unknown length in the small shape.

Example 22: (a) Prove that the figures given below are similar. (b) Given that the scale factor is 2, find the lengths of the two unknown sides s and t. Align the figures with corresponding sides and angles for comparison purpose.

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7F SIMILAR TRIANGLES

Similar triangle properties Two triangles are similar if one of the following conditions is identified:

1. All corresponding angles are equal (AAA).

2. All three corresponding pairs of sides are in the same ratio (SSS).

3. Two corresponding pairs of sides are in the same ratio and the included angles are equal (SAS).

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Example 23: For the similar triangles in the diagram below, find: (a) the scale factor. (b) the value of the pronumerals, x.

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Example 24: For the given triangles, find the value of the pronumerals, x.

Redraw and separate two distinct triangles.

Example 25: Find the height of the tree shown in the diagram below. Give the answer to 1 decimal place.

Redraw the two triangles separately then confirm their similarity using similar triangles properties. Through examining, the angles denotes by the symbol (( are equaled by corresponding angle property, while the angles denoted by the dots are also the same since both triangles shared the same angle. Therefore we can conclude that the two triangles are similar triangles.

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7G AREA AND VOLUME SCALE FACTORS

Area of similar figures If the lengths of similar figures are in the ratio of k, then the area of the similar shapes are in the ratio of k2. Example 26: For the two triangles shown below with the area of the large triangle equal to 100 cm2. Find the area of the small triangle, x cm2.

Example 27: For the two similar shapes shown, find the unknown length, x cm.

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Volume of similar figures If the lengths of similar figures are in the ratio k, then the volume of the similar shapes are in the ratio k3. Example 28: For the two similar figures shown, find the volume of the smaller cone if the volume of the large cone is 540 cm3.

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We can use the relationship between linear, area and volume scale factors to find any unknown in any pair of similar figures as long as a scale factor can be calculated.

Example 29: Find two similar triangular prisms with volumes of 64 m3 and 8 m3, find the total surface area of the larger triangular prism, if the smaller prism has a total surface area of 2.5 m2.

Linear Area Volume

k3

k2

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