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2.6 Line symmetry

Look at the shapes below. The symmetry of the shape on the left and its relationship to the shape on the right can be thought of in two ways:

  • Fold the left-hand shape along the central line. Then one side lies exactly on top of the other, and gives the shape on the right.

  • Imagine a mirror placed along the central dotted line. The reflection in the mirror gives the other half of the shape.

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Keep on learning

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There are more than 800 courses on OpenLearn for you to choose from on a range of subjects. 

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James Bamford on the National Security Agency

James Bamford talks to Nathan Thrall about the politics behind the Bush administration's evasion of the Foreign Intelligence Surveillance Act and the technology and scope of the National Security Agency's warrantless wiretapping program.


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1.7 Geological fieldwork

Although much can be learned from samples of rocks in the laboratory or at home, the ‘natural habitat’ of rocks is outdoors. Here the distribution and layout of different rocks is visible wherever rocks are exposed in places such as stream beds, cliffs, rocky shorelines, quarries, or road cuttings. The exposed rocks can be studied in just the same detail as individual laboratory samples, and geological fieldwork allows the size and extent of each rock unit to be seen and the relationships
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1.6 Interlude

Now that we have covered the features found in igneous, sedimentary and metamorphic rocks, and seen how these features can be explained by the processes that formed the rocks, here is a useful point at which to have a break before continuing with the next section. Before returning, you might like to see for yourself what types of rock you can find in your area. Can you identify their texture, or spot any fossils? Surfaces that haven't been obscured by grime or lichens are by far the best, as
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1.4.5 Fossils and ancient environments

An essential component of any environment is the plant and animal life that is adapted to the prevailing conditions. Fossil plants and animals are therefore wonderful sources of information about ancient environments. Plants can leave behind remains ranging from roots, leaves and twigs to seeds and pollen. Leaves and twigs are relatively fragile, and require a comparatively low energy environment (e.g. the mudflats of an estuary) for their preservation. Seeds, pollen and spores are surprising
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4.2 Earthquake magnitude

The magnitude of an earthquake is a measure of the amount of seismic energy released by it, so it is a quantitative scale. The scale of earthquake magnitude is called the Richter scale. Its development is described in Box 4, Charles Richter and the Richter earthquake magnitude scale. The Richter magnitude is calculated by first measuring the size of the largest ground motion recorded by a seismometer, a sensitive instrument that detects the ground movements
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Introduction to analysis
This free course is an introduction to analysis which looks at real numbers and their properties, with a particular emphasis on inequalities. Section 1 starts by revising rational numbers and their decimal representations. Then, real numbers are introduced as infinite decimals. Section 2 looks at rules for manipulating inequalities and finding the solution set of an inequality. Section 3 looks at various techniques for proving inequalities. Section 4 introduces the concept of a least upper bound
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Except for third party materials and otherwise stated (see http://www.open.ac.uk/conditions terms and conditions), this content is made available under a http://creativecommons.org/licenses/by-nc-sa/2

Medical statistics
This free course is concerned with some of the statistical methods used in epidemiology and more widely in medical statistics. Section 1 introduces cohort studies in which individuals are classified according to their exposure and followed forward in time to evaluate disease outcomes. Section 2 looks at models for cohort studies. Section 3 introduces case-control studies in which individuals are selected according to their disease status and past exposures are then ascertained. Section 4 covers
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3.3 Division

Division is probably the most awkward of the four arithmetic operations. Since you may have a calculator, you do not need to be able to carry out complicated divisions by hand, but you do need to carry out simple divisions in order to check your calculator calculations. Division is the reverse process of multiplication. The quantity 12 ÷ 3 tells us how many times 3 goes into 12. Since 4 × 3 = 12, 12 ÷ 3 = 4.

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2.6 Comparing measurements

In order to compare quantities, it is best to express them in the same units.

Example 10

Three children have just measured their own heights in metric units. Isaac says ‘My height is 1098’, Jasmine says ‘My height is 112’ and Kim says ‘Mine is 1.1’. What units were
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14 Practice dividing on paper

Now that you’ve learned the principles of doing division on paper, you may want to practise your new skills. If so, go to the Dividing decimals page of the math.com website and follow the instructions. You will need a pen and paper to carry out each calculation. You can then enter your answer on the website to check if it is correct.


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4.4 Summary of Part C

In Part C you have learnt that:

  • accurate law reporting allows for legal principles to be collated, identified and accessed;

  • there are many sources of law reports: Year Books (1275–1535), private reports (1535–1865), modern reports (1865 to present), the Law Reports, Weekly Law Reports, All England Law Reports, legal periodicals and newspapers, European Community Reports, DVD-ROMs and legal databases available via the internet.


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3.4 Cylinders and shapes with a uniform cross-section

An important idea when calculating volumes of simple shapes is that of a cross-section. In the case of the rectangular box considered above, it is possible to slice through the box horizontally so that the sliced area is exactly the same as the area of the base or top; in other words, the areas of the horizontal cross-sections are equal.

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2.8 The angles of a triangle

The sum of the angles of any triangle is 180°. This property can be demonstrated in several ways. One way is to draw a triangle on a piece of paper, mark each angle with a different symbol, and then cut out the angles and arrange them side by side, touching one another as illustrated.


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Try some yourself

Question 1

Draw a line of symmetry on each of the shapes below.

2.7 Rotational symmetry

There is another kind of symmetry which is often used in designs. It can be seen, for instance, in a car wheel trim.

Look at the trim on the left. It does not have line symmetry but
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1.4 Parallel lines

Two straight lines that do not intersect, no matter how far they are extended, are said to be parallel. Arrows are used to indicate parallel lines.


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1.3.4 Vertically opposite angles

When two straight lines cross, they form four angles. In the diagram below, these angles are labelled α, β, θ and φ and referred to as alpha, beta, theta and phi. The angles opposite each other are equal. They are called vertically opposite angles. Here α and β are a pair of vertically opposite angles, as are θ and φ. Although such angles are called ‘vertically opposite’, they do not need to be vertically above and bel
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Try some yourself

Question 1

A company carried out a survey, recording how staff in a particular office spent their working time. The table shows the average number of minutes spent in each hour on various activities.

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