Operacion mundo: Mathematics B 4. Secondary DF (muestra)
DUAL FOCUS
4
SECONDARY EDUCATION
MATHEMATICS b
ANDALUSIA
INDEX
Train yourself by solving problems......... 7
Summary • Problems
1. Real numbers
9
Summary • 1 Irrational numbers • 2 Real numbers: the real number line • 3 Sections of the real number line: intervals and half-lines • 4 Roots and radicals • 5 Approximate numbers. Errors • 6 Numbers in scientific notation. Error control • 7 Logarithms • The final challenge
2. Polynomials and algebraic fractions 21
Summary • 1 Polynomials. Operations • 2 Ruffini’s rule • 3 Root of a polynomial. Finding roots • 4 Factorising polynomials • 5 Divisibility of polynomials • 6 Algebraic fractions • The final challenge
3. Equations, inequations and systems 29
Summary • 1 Equations • 2 Systems of equations • 3 Inequations with one unknown • 4 Linear inequations with two unknowns • The final challenge
4. Similarity. Applications
41
Summary • 1 Similarity • 2 Homothety • 3 Rectangles with interesting dimensions • 4 Similarity between triangles • 5 Similarity between right-angled triangles • 6 Similarity of right-angled triangles in threedimensional shapes • The final challenge
5. Trigonometry
51
Summary • 1 Trigonometric ratios of an acute angle • 2 Basic trigonometric identities • 3 Using a calculator in trigonometry • 4 Trigonometric ratios of 0° to 360° • 5 Angles of any size. Trigonometric ratios • 6 Solving right-angles triangles • 7 Solving non right-angles triangles • 8 Interesting theorems • The final challenge
6. Analytic geometry
61
Summary • 1 Vectors in the plane • 2 Calculations using vectors • 3 Vectors which represent points • 4 Midpoint of a segment • 5 Aligned points • 6 Equations of a straight line • 7 Straight lines. Parallelism and perpendicularity • 8 Straight lines parallel to the coordinate axes • 9 Relative positions of two straight lines • 10 Distance between two points • 11 Equation of a circumference • 12 Studying motion • The final challenge
7. Functions I .....................................................
71
Summary • 1 Basic concepts • 2 How functions are represented • 3 Domain of definition • 4 X- and Y- intercepts. Sign of a function • 5 Continuous functions. Discontinuities • 6 Variations in a function • 7 Tendency and periodicity • 8 Linear functions • 9 Quadratic functions • The final challenge
Summary • 1 Statistics and statistical methods • 2 Frequency tables • 3 Statistical parameters: x –and σ • 4 Measures of position for isolated data • 5 Measures of position for grouped data • 6 Box plots • 7 Statistical inference • 8 Statistics in the media • The final challenge
10. Bivariate distributions
103
Summary • 1 Bivariate distributions • 2 Correlation value • 3 Using the line of best fit to make estimations • 4 Does correlation imply cause and effect? • 5 Bivariate distributions with a calculator • The final challenge
11. Combinatorics
111
Summary • 1 Product-based strategies • 2 Variations and permutations (the order matters) • 3 When the order does not matter. Combination • 4 An interesting number triangle • 5 Newton’s binomial • The final challenge
12. Calculating probabilities
119
Summary • 1 Random events • 2 Event probability. Properties • 3 Probability in simple experiments • 4 Probability in compound experiments • 5 Compound dependent events • 6 Compound independent events • 7 Contingency tables • The final challenge
TAKE ACTION!
1 REAL NUMBERS
REAL NUMBERS
Irrational numbers
The real number line
Roots and radicals
Approximate numbers
Operations
Rationalising the denominator
Representing
Intervals and half-lines
Union
Representation
Intersection
FOCUS ON ENGLISH
Since the time of Archimedes (3rd century BC), we have known fairly accurate approximations of one of the most famous irrational numbers in the history of mathematics: the number π
Take on ‘The final challenge’ and write a report on the approximations to the number π that you have obtained.
Scientific notation Logarithms
Operations
Absolute error
Telative error
Fractional numbers
Radicals
Real numbers
Decimal
Napierian
Properties
Listening
Listen and repeat. The vocabulary is at anayaeducacion.es.
Discover
In ancient times, mathematicians had already realised that the relationship between the perimeter of a circle (the length of its circumference) and its diameter was always the same, regardless of the size of the circle. In the 17th century, this relationship was expressed for the first time as the number ‘pi’, the first letter of the Greek word periphereia (perimeter).
Reading
Read the text ‘Rational and irrational numbers’ at anayaeducacion.es and answer the questions there.
Irrational numbers 1
Rational numbers are numbers that can be expressed as the quotient of two integers. Their decimal expression is terminating or recurring.
Irrational numbers are non-rational numbers, in other words, they cannot be expressed as the quotient of two integers. Their decimal expression is infinite. There are infinite irrational numbers.
The diagonal of a square: the number √ 2
The Pythagorean theorem gives us the value of the diagonal of a square with sides measuring 1:
d = 11 2 22+=
We are going to demonstrate that 2 is irrational, in other words, that it cannot be expressed as the quotient of two integers. We will do this through reduction to absurdity (supposing it is rational and then showing it reaches an absurdity).
— Suppose that 2 is rational. In this case: b a 2=
— We square the two sides: 2 = b a 2 2 → a 2 = 2b 2
Since b 2 is a perfect square, it contains the factor 2 an even number of times. Therefore, 2b 2 has factor 2 an odd number of times, which is impossible since 2b 2 = a 2 is another perfect square. Therefore 2 is not rational.
As with 2 , if p is not a perfect square, p is irrational.
If p is not an exact nth power, p n is an irrational number.
The result of performing an operation using a rational number and an irrational one is irrational (except in the case of multiplying by zero).
The golden ratio: Φ = √5 + 1 2
The diagonal of a pentagon with a side length of 1 unit is ( 5 + 1) : 2 which is obviously irrational. The golden ratio is also known as the golden number. d U = d l
The number π
2r L r L 2 r =
The number e
π is the relationship between the length of the circumference of a circle and its diameter. It is an irrational number and, therefore, has infinite non-recurring decimal digits.
The number e is irrational. Its approximate value is 2.7182... and you will come across it in many different situations, for example in the exponential function describing the growth of an animal or plant population.
Observe
When we break down a perfect square into prime factors, each prime number appears an even number of times. For example:
All the exponents are even.
1 Demonstrate that the following numbers are irrational: a) 3 b) 5 + 4 3
2 Look at the following numbers: –2; 1.7; 3 ; . 42 ! ; –. 37 5 ! ; 3π; –2 5 ; 5e
Find the rational numbers and express them as a quotient of two integers.
3 Writing. Demonstrate that the golden number, Φ , is irrational.
Real numbers: the real number line 2
The set formed by the rational and irrational numbers is called the set of real numbers and is represented by the symbol Á.
With real numbers, we can perform the same operations as with rational numbers: addition, subtraction, multiplication and division (except for zero), and the same properties are maintained.
We can also extract roots of any index (except roots with even indices of negative numbers) and the result is still a real number. This does not happen with rational numbers.
We can see that the N , Z , Q and, now, Á sets are closed when working with addition and multiplication; in other words, both the sum and product of two elements of one of these sets are also part of that set.
The real number line
If we place an initial point (0, zero) on a line and mark the length of one unit, each point on the line will correspond to a rational or irrational number. In other words, each point on the line corresponds to a real number. This is why we call this number line the real number line.
Between any two real numbers, no matter how close together they are, there are infinite rational and irrational numbers.
1 Speaking. Look at the following numbers:
Observe
This is not the case for negative integers, fractions and irrational numbers. For example:
a) Classify into rational and irrational numbers.
b) Put them in order from smallest to largest.
2 Place the following numbers on a diagram like the one on the right: 1; 723 # ; 1 – 2 ; 3.5; 9 11 ; 4 1 ; 6 ; 4 r ; –104
Representing numbers on the real number line
• Using Thales’ theorem to represent fractional numbers
We can use Thales’ theorem to place the number 5 14 on the real number line: 5 14 = 2 + 5 4
0 4 2 + 5 1 2 3
• Using the pythagorean theorem to represent radicals
We can use the following procedure to calculate n for any n ∈ N: For
• Approximate representation of real numbers
We can represent a real number given as a decimal with as close an approximation as we like. For example: 842 5 = 3.8464…:
• Notice how each expansion involves splitting the previous subinterval into ten parts and then taking one of those parts. We can get as close to the number we want as we like.
Depending on their type, real numbers can be represented on the real number line either exactly or with as close an approximation as we like.
3 Writing. a) Demonstrate that the point represented is 21 √21
0 1
b) Represent 27 (27 = 36 – 9) and 40 (40 = 36 + 4).
Observe
However, most real numbers cannot be represented exactly using this type of procedure. We normally use an approximate representation.
4 Speaking. What number is the arrow in the following diagram pointing at?
Represent 2.716 in the same way.
Sections of the real number line: intervals and half-lines 3
• The open interval (a, b) is the set of all the numbers between a and b, but not including a or b : { x ∈ Á/ a < x < b }.
• The closed interval [a, b] is the set of all the numbers between a and b, including both of them: { x ∈ Á/ a ≤ x ≤ b }.
• The half-open interval (a, b] is the set of all the numbers between a and b, including b but not a : {x ∈ Á/ a < x ≤ b }.
• The half-open interval [a, b) is the set of all the numbers between a and b, including a but not b : {x ∈ Á/ a ≤ x < b }.
Half-lines and the real number line
• – ∞, a) are the numbers less than a {x ∈ Á/ x < a}
• (– ∞, a] are the numbers less than a and a itself: {x ∈ Á/ x ≤ a}
• (a, +∞) are the numbers greater than a: {x ∈ Á/ x > a}
• [a, +∞) are the numbers greater than a and a itself: {x ∈ Á/ x ≥ a}
The real number line itself is represented as an interval like this: Á = (– ∞, +∞).
When there is no number that satisfies a specific condition, it is represented by the empty set, whose symbol is ∅. For example, the values of x for which x 2 < 0 correspond to the empty set: {x ∈ Á / x 2 < 0} = ∅
Draw these intervals and inequalities:
• Half-open interval (2, 3] 2 3
• Half-line (0, +∞) 0
• {x ∈ Á/ –2 ≤ x ≤ 0} –2 0
• {x ∈ Á/ 0 < x < 1} 0 1 a a a a
1 Reading. Write the following sets in interval form and represent the numbers that meet the conditions given on the number line:
a) Numbers between 5 and 6, both included.
b) Greater than 7.
c) Less than or equal to –5.
Observe
The union of two intervals or half-lines is represented by ∪: (–∞, 2) ∪ (0, 5] = (–∞, 5]
The intersection of two intervals or half-lines is represented by ∩: (–∞, 2) ∩ (0, 5] = (0, 2)
2 Write the following in interval form and draw them: a) {x ∈ Á/ 3 ≤ x < 5} b) {x ∈ Á/ x ≥ 0} c) {x ∈ Á/ –3 < x < 1} d) {x ∈ Á/ x < 8}
3 Write as inequalities and draw them. a) (–1, 4] b) [0, 6] c) (–∞, – 4) d) [9, +∞)
Roots and radicals 4
The nth root of a number, a, (written a n ) is the name given to a number, b, that meets the following condition: a n = b if b n = a a n is called a radical; a, a radicand; and n, the index of the root.
Radicals in exponential form
Radicals can be expressed as powers:
Operations with radicals
• Simplification of radicals
When expressing radicals as powers, we sometimes notice that they can be simplified. 93 33 3 // 4 2 4 24 12 = === We have applied property ❶.
• Reduction of radicals to a common index
To compare two radicals with different indeces, we reduce them to a common denominator.
To compare 586 3 to 70 ,
We have applied property ❶.
• Taking out factors from roots
In order to simplify some radicals, and to add and subtract them, sometimes we need to take out factors from a root.
··
• Product and quotient of radicals with the same index ·
• Simplification of products and quotients of radicals
We have applied properties ❶
Observe
• If a ≥ 0, a n exists no matter what the value of n is.
• If a < 0, only its roots of odd indices exist.
• In general, a positive number, a, has two square roots: a and – a . 1
• Power of a radical
We have applied property ❹.
• Root of a radical
22 3 6 = 55 3 4 12 = We have applied property ❺.
• Addition and subtraction of radicals
Only identical radicals can be added together. Sometimes, the possibility of simplifying an addition of radicals is hidden. First, we need to take out any factors that we can from the roots or simplify them.
Rationalising the denominator
The process we use to remove the radicals from the denominator is called the rationalisation of the denominator.
In each case, we should ask ourselves this question: Which expression do I need to multiply the denominator by for the product not to have radicals? Once we have found the expression, we will also multiply the numerator by it so that the final result does not vary.
1st case: square roots. For example:
2nd case: other roots. For example:
3rdcase: addition and subtraction of roots. For example:
1 Express the following in exponential form:
2 Which of the two is greater in each case?
31 4 and 13 3
3 Reduce.
Simplify
Calculate.
• We call the expression ab – the conjugate of ab +
• And, the other way round, ab + is the conjugate of ab –
6 Rationalise the following denominators:
Approximate numbers. Errors 5
Approximations and errors
We call the figures used to express an approximate number significant figures. We only need to use those we are sure of, and in such a way that they are relevant for the information we want to transmit.
The absolute error of an approximate measurement is the difference between the real value and the approximate value.
Absolute error = |Real value – Approximate value|
The real value is generally unknown. Therefore, the absolute error is also unknown. The most important thing is to be able to put limits on it: the absolute error is less than… We get the limit of the absolute error from the last significant figure used.
The relative error is the quotient of the absolute error and the real value. It is therefore lower the more significant figures we use. The relative error is usually also expressed as a percentage (%).
If the capacity of a swimming pool is 718 900 L, it might be more sensible to say it was 719 m3, using only three significant figures. However, if the measurement was not very accurate, or we did not want to give such an exact measurement, it might be better to say 720 m3, or even 72 tens of m3.
If we say that the capacity of the swimming pool is: 719 m3, the last significant figure (9) gives units of m3. The absolute error is less than half a cubic metre (error < 0.5 m3).
The relative error is less than . 719 05 < 0.0007 = 0.07 %.
1 Speaking. True or false? Explain your answers.
a) Saying that a swimming pool holds 147 253 892 thousand drops of water is correct if the measurements are very accurate.
b) If we correctly estimate that the number of drops of water that a swimming pool holds is 15 ten
Observe
a) 34 m has 2 significant figures.
b) 0.0863 hm3 has 3 significant figures.
c) It is possible that 53 000 L only has 2 significant figures if the zeros are just used to designate the number. In this case, it would be better to say 53 thousands of litres.
thousands of millions, the absolute error is less than half of ten thousand million drops; in other words, absolute error < 5 000 000 000 drops.
c) If the relative error for a certain measurement is less than 0.019, we can say that it is less than 19 %.
d) If the relative error for a certain measurement is less than 0.019, we can say that it is less than 2 %.
Numbers in scientific notation. Error control
— Are written as two factors: a decimal number and a power of 10.
— The decimal number is greater than or equal to 1 and less than 10.
— The power of 10 has an integer exponent.
The numbers 3.845 · 1015 and 9.8 · 10–11 are in scientific notation.
3.845 · 1015 = 3845000000000000, is a ‘large’ number.
This form of expression is very useful for handling very large or very small approximate quantities, since:
• You can see the ‘size’ of the number at a glance. It can be seen in the second factor and it is given by the exponent of 10.
• You can see how accurate the quantity is. The more significant figures given in the first factor, the more accurate the number is.
Operations with numbers given in scientific notation
• Product and quotient
We do the operations involving the decimal components separately from the powers of 10. Then, we adjust the result so that it is expressed correctly using scientific notation.
We prepare the addends so that they have the same power of base 10, which we use as a common factor. Then we readjust the result.
3.7 · 1011 + 5.83 ·
Controlling error in a number in scientific notation
If we are told that ‘there are 2 500 bags of flour in this warehouse’, this could be an approximate quantity. If we use scientific notation, the expression is unambiguous: 2.5 · 102 means that there are only two significant figures. And if there are three, we write 2.50 · 102.
Remember
We can easily do operations in scientific notation with a calculator. To write a number in scientific notation you use the � key. For example:
1 Calculate and check your answers with a calculator:
a) (6.4 · 105) · (5.2 · 10– 6)
b) (2.52 · 104) : (4 · 10– 6)
c) 7.92 · 106 + 3.58 · 107
d) 8.113 · 1012 – 8 · 1011
2 The distance from the Earth to the Sun is 149 000 000 km.
a) Express it using scientific notation.
b) Express it in centimetres to two significant figures.
c) Give a limit of the absolute and relative errors for the three cases mentioned above.
Logarithms
We call the exponent that base a has to be raised to in order to obtain P (a > 0 and a ≠ 1) base-a logarithm of P. It is written loga P. log a P = x ⇔ a x = P
Find the value of these logarithms:
a) log6 1 296 b) log2 0.125
a) 1 296 = 64. Therefore, log6 1 296 = 4.
b) 0.125 = 1
Properties of logarithms
1. Two simple logarithms
2–3. Therefore, log2 0.125 = –3.
The logarithm of the base is 1. The logarithm of 1 is 0 in any base.
log a a = 1 log a 1 = 0
2. Product and quotient
The logarithm of a product is the sum of the logarithms of the factors. The logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.
log a (P · Q ) = log a P + log a Q log a Q P = log a P – log a Q
3. Power and root
The logarithm of a power (P k or PP / n n 1 = ) is equal to the exponent multiplied by the logarithm of the base of the power.
a P k = k
4. Base change
If we know how to calculate base-a logarithms, we can use the formula on the right to calculate logarithms with any base, b
log b P = log log b P a a
Decimal logarithms
Base-10 logarithms are called decimal logarithms. For a long time, they were the most widely used logarithms. This is why we just write log, without the base.
For example, log 10 = 1, log 100 = 2, log 1 000 = 3, log 0,0001 = – 4.
And log 587 = 2,… since 587 is greater than 100 but less than 1 000.
There is a specific button for these logarithms on calculators. But in modern calculators, you have to press the SHIFT button to access this function:
log 200 → s 200 = 2.301029996
Napierian logarithms
Logarithms with base e are called Napierian logarithms and they are written ln (in other words, loge x = ln x).
On calculators there is a button, , for accessing these logarithms.
Using a calculator
The calculators we use today have three keys for calculating logarithms: , j and . The first two are easy to find. The third (decimal logarithm), however, is more complicated. It is the second function of the button, so to use it, we have to press s .
To find a logarithm with a base other than 10 and e, newer calculators also include the j button. However, sometimes it may be preferable to use property 4, base change (see page 16), rather than this button. To do this, it is better to use the ln key, since that way you do not need to use the shift button.
• log 5 2 → jí 2 ””5 = 4.64385619
• log 5 2 = ln ln 2 5 → l 5 )/lí 2 = 4.64385619
1 Use the definition above to find these logarithms:
3 Writing. Use the calculator to find log 7 and log 70 and explain why both have the same decimal part.
4 Apply the definition of a logarithm and calculate. log4 163 + log4 2 + log 0.0001 + log 100 10 3
5 If log x = 1.3 and log y = 0.8, calculate: a) log (x · y) b) log () xy
c) log x y 2 d) log y x
The final challenge
Real numbers
TAKE ACTION!
Since the time of Archimedes (3rd century BC), we have known fairly accurate approximations of one of the most famous irrational numbers in the history of mathematics: the number π Take on ‘The final challenge’ and write a report on the approximations to the number π that you have obtained.
Archimedes was the first mathematician in history to devise a procedure for calculating the value of π
It consisted of inscribing and circumscribing a polygon in a circle of radius 1 and bounding the value of π between its perimeters.
Perimeter circumscribed polygon
Perimeter inscribed polygon < 2π <
First, he used a hexagon. Then, he repeated the process by multiplying the number of sides by 2: 12, 24, 48 and 96. Thus, at each step, the approximation became more accurate.
In this way, he determined that 71 10 < π < 7 22 . Interesting, isn’t it?
1 Is the number π rational or irrational? Explain your answer.
2 Graph and express the intervals or inequalities. 1.6 ≤ x ≤ 1.8 3.1 < x < 3.2 2.1 < x ≤ 2.5 –2 < x < 0 x ≥ 3.5
2.7 ≤ x < 2.8
a) In which of the above intervals does the number π lie?
b) What about the number e?
c) What about the number Φ?
Final product
What values did you obtain? Do you think they are good approximations to the number π? Which of the approximations do you think is the best? Write a
3 In addition to Archimedes’ method, there are other ways to calculate the value of the number π . Here are two:
a) Monte Carlo method.
Search on the Internet what this method is and write a short summary.
b) Numerical series.
(General term: n 1 2 n. ,, , ·· · 13 88 88 57 9111315 · , approaches π.
(General term: ()nn 2 8 + , n = 1, 5, 9, 13…).
Choose one of these series and calculate with GeoGebra the sum of its terms to check that they are close to π.
report with the difficulties you encountered trying to implement each method. Which one do you think is more effective?
4 SIMILARITY. APPLICATIONS
Ratio of similarity
Maps, plans, models
Get together with other classmates and go out and measure the height of a local landmark using the tools provided in 'The final challenge'. Did you get good approximations of the real values with your calculations?
SIMILARITY. APPLICATIONS
Similar shapes
Relationship between areas
Relationship between volume
Scale
Homothety
Applications
Homothety in threedimensional space.
Similar rectangles
Reductions and enlargements
Similar rectangles
Similarity between triangles
Thales’ theorem
Golden rectangle
Triangles in Thales’ position
Similarity criteria
Right-angled triangles
Leg rule
Height rule
FOCUS ON ENGLISH
Listening
Listen and repeat. The vocabulary is at anayaeducacion.es
Discover
Thales of Miletus (6th century BCE) was a famous Greek mathematician. He visited Egypt in order to broaden his knowledge. According to legend, while he was there he measured the height of the pyramid of Cheops. To do this, he identified the time of day when the length of his shadow was the same as his height. He then applied this to the pyramid to determine its height.
Reading
Read the text 'An important historical figure' at anayaeducacion.es and answer the questions.
Similarity 1
In two similar figures:
• The corresponding angles are equal.
• The lengths of the corresponding segments are proportional. The ratio of proportionality is called the ratio of similarity.
Similar shapes in everyday life
A scale is the quotient between each length in the reproduction (map, plan, model) and the corresponding length in reality. This means it is the ratio of similarity between the reproduction and reality.
A scale of 1:200 means, as you already know, that 1 cm on the plan corresponds to 200 cm = 2 m in reality. A scale of 100:1 means that 1 cm in reality corresponds to 100 cm = 1 m on the diagram or model. Scales like this are used to represent very small objects.
Relationship between areas and volumes
If the ratio of similarity between two figures is k, the ratio of their areas is k 2 and the ratio of their volumes is k 3 .
The ratio of the areas of two similar figures is equal to the square of their ratio of similarity.
The ratio of the volumes of two similar figures is equal to the cube of their ratio of similarity.
If a model has a scale of 1:200, the ratio between the area of a plot of land in real life and in the model is 2002 = 40 000. And the ratio between the volume of a building in real life and in the model is 2003 = 8000000.
1 a ) One of the buildings in the model above is a cuboid. The dimensions of its base are 9 cm × 6.4 cm and it is 4 cm tall. Find its dimensions, the area of its facade and its volume in reality.
b) The area of a football field in the model is 32 cm2. What is its actual area?
c) One of the houses in the model is made of 0.3 cm3 of polystyrene. What is its volume in reality?
2 Reading. At the Atocha station in Madrid there is a work by the famous painter and sculptor Antonio López: it is a newborn baby’s head with a perimeter of 7 m. If the average head circumference of a baby at birth is 35 cm, calculate:
a) The scale used by the sculptor.
b) If the sculpture has a volume of 5.8 m3, what would the volume of the baby’s head be in real life?
Homothety 2
In the transformation on the right, each point from the blue shape (for example, A) has been transformed into a point from the red shape (A' ) according to the following conditions: O, A and A' form a straight line and · ' OOAA 2 =
This is a homothety of centre O and with a ratio of 2.
A homothety of centre O and ratio k is a transformation in which each point, P, corresponds to another point, P' , such that:
• O, P and P' form a straight line.
• k = OP OP l if k > 0 if k < 0 P O P' O P P'
Two homothetic shapes are similar and their ratio of similarity is k. Their corresponding sides will also be parallel.
Applications of homothety
• Similar rectangles
Note how we can demonstrate that two rectangles are similar and construct a rectangle that it is similar to another given rectangle:
proof that the rectangles t and t' are similar
• Reductions and enlargements of a shape
Halving (in red) and doubling (in blue).
Homothety in three-dimensional space
Two shapes that are homothetic in three-dimensional space are also similar. In addition, their corresponding edges and faces are parallel. The inverse is also true: if two shapes are similar and their corresponding elements (faces, edges, etc.) are parallel, then there is a homothetic transformation that will convert one into the other.
construction of a rectangle similar to a National ID document
Homothetic centre at one of the vertices.
Homothetic centre at a point inside the shapes.
1 Speaking. Where is the homothetic centre of the Sun and Moon during a total solar eclipse?
2 Listening. Why are there total and annular eclipses? Explain your answer at anayaeducacion.es.
Rectangles with interesting dimensions 3
There are two rectangular objects you use all the time that are closely linked to the concept of similarity: sheets of A4 paper and your ID document. We can use the following property to show that these rectangles are similar:
Two rectangles are similar if their dimensions (length and width) are proportional.
A sheet of A4 paper
The most commonly used sheets of paper (A4) have a curious property: if we cut them in half, each of the pieces is similar to the initial sheet. We can use this property as a basis for finding the ratio between their sides
We take the shorter sides as the unit. The similarity between both rectangles gives us the following ratio:
Observe
A sheet of paper that is double the size of an A4 sheet is called A3.
Equally, paper double the size of A3 is A2, and double A2 is A1.
Half A4 is A5.
these two rectangles are similar
Therefore, for a rectangle and a half of the same rectangle to be similar, the ratio between their sides must be 2 .
• Proof by folding sheets of paper
We can prove that a sheet with the dimensions a × b is A4 or similar by making two simple folds:
Half A5 is A6.
These are all similar.
1 Given that the dimensions of a sheet of A4 paper are 210 mm × 297 mm, calculate the dimensions of the following sheets of paper: A0, A5 and A8.
2 a) Calculate the series of areas of these sheets of paper: A1, A2, A3…
b) Find the sum of the infinite terms and show that, according to the diagram on the right, this sum must be twice the first term, or A0.
Golden rectangles
The rectangle in the margin has an unusual property: if we cut a square off it, the rectangle that is left is similar to the initial one. Let’s see what the relationship between its dimensions are:
Taking the shorter side as the unit, the similarity of the rectangles gives us the following ratio: s s 11 1 –= s s – 1 1 1
The relationship between the sides is s = 2 15 + . It is the golden ratio, Φ
Therefore, this rectangle and all rectangles that are similar to it are called golden rectangles.
Spanish national ID documents and credit cards are examples of golden rectangles. Golden rectangles are also used frequently in architecture.
We can use a ruler and compass to check that a rectangle is golden, using a method based on the construction of the number Φ.
1 We use the shorter side, 1, as the radius of an arc ending on the longer side: OA = 1.
2 We draw the line AP = 1, forming a square.
3 Q is the midpoint of AO. And PQ 2 5 =
4 We use PQ as the radius of an arc ending on the long side of the rectangle. BO measures: 2
3 Reading. If you cut a 2.5 cm strip off the long side of an A4 sheet, you get a golden rectangle. Create two.
Cut a square off one of them.
Check that the remaining rectangle is similar to the initial rectangle.
4 If we add a square to an A4 sheet, the resulting rectangle, which we will call A4 plus, has the following property: if we remove two squares from it, the remaining rectangle is similar to the initial one.
A4 plus
a) Check this with an A4 sheet.
b) Prove it, bearing in mind that the dimensions of the A4 plus are 2 + 1 and 1, and those of the excess rectangle are 1 and 2 – 1.
Similarity between triangles 4
Thales’ theorem
If a, b and c are parallel lines and they intersect with another two straight lines, r and s, then the segments between them will be proportional.
= As a result, we can say that:
Conversely, if the segments AB and BC are proportional to AB'' and BC'' and the lines a and b are parallel, then line c is parallel to them.
Thales’ theorem can be used to study the similarity of triangles.
Similar triangles
Two similar triangles have:
• Proportional sides: of ' ' ' a a b b c c ratiosimilitary == =
• Equal angles:
Triangles in Thales’ position
Triangles ABC and AB'C' on the right have a common angle, A ^ . In other words, the smaller triangle fits into the larger one. In addition, the sides opposite A ^ are parallel. We say that these two triangles are in Thales’ position.
Two triangles in Thales’ position are similar.
1 Identify the triangles in Thales’ position in each figure and calculate the length of the segment DE in each:
Criteria for similar triangles
In order to know whether two triangles are similar, we do not need to check that all the conditions mentioned on the previous page are met. We only need to check some of them.
The set of conditions that allow us to be sure that two triangles are similar are called similarity criteria. If the triangles fulfil these conditions, we can be sure that they are similar triangles.
• First criterion
Two triangles are similar if they have two pairs of equal angles.
ABC & is similar to '' ' AB C & if:
and B
• Second criterion
Two triangles are similar if their sides are proportional.
ABC & is similar to '' ' AB C & if:
• Third criterion
Two triangles are similar if they have an equal angle and the sides that form it are proportional.
ABC & is similar to '' ' AB C & if:
2 Writing. We are at point A . We want to calculate the distance to a remote, inaccessible location, C . To do so, we plot another point, B , and measure: AB = 53 m. We also measure the angles A ^ = 46° and B ^ = 118°.
We draw triangle A'B'C' with the following measurements: AB'' = 53 mm, A ^ ' = 46°, B ^ ' = 118°.
a) Draw triangle A'B'C' in your notebook.
b) Explain why '' ' AB C & is similar to ABC &
c) Measure AC'' with a ruler.
d) Work out the distance to C, AC .
Similarity between right-angled triangles 5
Similarity criterion for right-angled triangles
Two right-angled triangles are similar if one of the acute angles of one triangle is equal to one of the acute angles of the other triangle.
How do we know? Well, their right angles are equal, so if they have another equal pair of angles, their third angles also have to be equal.
Consequences of this similarity criterion
All triangles obtained by drawing lines perpendicular to one of the sides of an angle are similar.
All the triangles in figure ❶ (ABO, A'B'O, A''B''O) are similar because they have the angle α in common. Therefore, thanks to the above criterion, we know that their sides are proportional.
In a right-angled triangle, the height from the hypotenuse creates two triangles that are similar to the original.
The figure ❷ below shows three right-angled triangles: ABC, AMB and AMC.
— ABC and AMB are similar because they share angle B ^ .
— ABC and AMC are similar because they share angle C ^ .
Leg rule
As we have just seen, each of the triangles formed by drawing the height from the hypotenuse of a right-angled triangle is similar to that triangle.
• As the triangles ABC and MAC are similar: b a m b = → b 2 = a · m
• The similarity of ABC and MBA gives us: c a n c = → c 2 = a · n
The square of a leg is equal to the product of the hypotenuse and the projection of that leg on the hypotenuse. In other words: b 2 = a · m and c 2 = a · n
1 In each of the following right-angled triangles the height BH has been drawn from the hypotenuse. In each case, find the segments x and y .
Height rule
As the triangles MAC and MBA are similar: m n h h = → h2 = m · n
The square of the height from the hypotenuse is equal to the product of the two segments created by the division of the hypotenuse. In other words:
2 For this right-angled triangle, calculate the lengths h, m and n.
Similarity of right-angled triangles in three-dimensional shapes
used in three-dimensional shapes.
Find the volume of a truncated cone that is 9 cm high, where the radii of its bases are 20 cm and 8 cm.
We call the increase in the height x. It is important to bear in mind the similarity of both triangles: the small one, with legs 8 and x ; and the large one, with legs 20 and x + 9:
The volume of the truncated cone is the difference between the volumes of the two cones:
1 Writing. Calculate the volume of a truncated cone with a height of 9 cm, where the radii of the bases are 20 cm and 35 cm.
a) Do it step by step, giving your reasoning.
b) Check it by applying the above formula.
2 Calculate the volume of a truncated regular square pyramid with square bases.
Sides of the squares: 40 cm and 16 cm
Height: 9 cm
The final challenge
TAKE ACTION!
SIMILARITY. APPLICATIONS
Get together with other classmates and go out and measure the height of a local landmark using the tools provided in 'The final challenge'. Did you get good approximations of the real values with your calculations?
This weekend Marta and some of her friends visited a famous park near the town where they live. Besides being a large green and leisure area, they know that inside the park they can find replicas of 17 European monuments such as the Eiffel Tower (Paris), the Belém Tower (Lisbon), the Atomium (Brussels) or the Puerta de Alcalá (Madrid), among others.
1 The monuments in the park are built to scale.
a) The Eiffel Tower in Paris is 324 m high. If the replica is built to a 1:10 scale, what is its height?
b) The Belém Tower in the park is 11.67 m high. If it is built to a 1:3 scale, what is the height of the original?
c) The Atomium represents an iron atom magnified 165 billion times. The structure is 102 m high and consists of 9 spheres of 18 m in diameter connected by tubes with escalators. If the replica of the park is made at a 1:9 scale, what is the surface area of one of the spheres? And its volume?
RemembeR
3 Marta and her friends have visited the replicas of the monuments at points A and B . When they have finished visiting the monument at point C , they want to return to the meeting point, P , by the shortest route. When they have finished visiting the monument at point C they want to return to the meeting point, P , by the shortest route. How far will they have to walk?
Final product
Surely in your town or neighbourhood there is a representative monument. Choose some of them among several classmates and calculate their height using your own and some other reference object.
2 The replica of the Puerta de Alcalá is a little smaller than the original. Marta has stood in front of it and drawn a diagram to calculate its height.
a) If Marta is 1.6 m tall and the street lamp is 3.6 m tall, what is the height of the replica of the Puerta de Alcalá?
b) If the replica is at a scale of 1:1.1, how big is the original Puerta de Alcalá?
Look up the real height of the monument on the Internet and assess whether the results you have obtained are good approximations. In case of error, reflect on the possible causes.
REAL NUMBERS 1
Irrational numbers
Rational number:
It be expressed as the quotient of two integers. Its decimal expression is terminating or recurring.
Irrational number:
It cannot be expressed as a quotient of two integers. Its decimal expression is infinite and non-recurring.
Some irrational numbers
The number e e = 2.7182… p n is irrational if p is not an exact nth power.
1 Write three irrational numbers other than Φ , π and e
Real numbers
2 Write three real numbers between 4.547 and 4.548.
Intervals
Half-lines
3 Write the following set in interval form: ‘all real numbers less than 5 and greater than or equal to –1’.
4 Write two inequalities whose intersection is the interval (3, 7].
Operations with radicals
Rationalising the denominator
1 Square roots 2 Other roots 3 Addition and subtraction of roots
Approximations and errors
Absolute error = |Real value – Approximate value|
5 What is the necessary condition for two radicals to be added and subtracted without using decimal numbers?
6 What are the absolute and relative errors when approximating 6.33... by 6.3?
Power of base 10 (with integer exponent)
a. b c d... . 10n
Decimal part
Integer part (a single figure)
Logarithms
Properties
• log a a = 1
• log a 1 = 0
7 Why is a number like 3 500 not usually written in scientific notation? 8 Relate the numbers 2; 0.125 and –3 using logarithms and powers.
• log a (P · Q ) = log a P + log a Q
log a Q P = log a P – log a Q • log a P k = k log a P
log a log P n P 1 n a =
SIMILARITY. APPLICATIONS 4
Similar shapes
In two similar shapes:
• The corresponding angles in similar shapes are equal.
• The lengths of the corresponding segments are proportional. The ratio of proportionality is called the ratio of similarity.
The scale is the ratio of similarity between the reproduction and reality.
1 cm on the plan is 200 cm in reality. 1:200
If the ratio of similarity between two shapes is k, then the ratio of their areas is k 2 .
If the ratio of similarity between two three-dimensional shapes is k, then the ratio of their volumes is k3
Homothety
• A homothety of centre O and ratio k is a transformation in which each point, P, corresponds to another point, P' , such that:
• O, P and P' form a straight line.
•
Homothety in three-dimensional space
1 Write a scale that corresponds to an enlargement.
Applications
• Similar rectangles (those whose dimensions are proportional)
• Reductions and enlargements of a shape
Homothetic centre at one of the vertices.
2 What is the result of applying a homotopy of centre O and ratio 2 to the square of vertices O , (1, 0), (0, 1), (1, 1)?
Homothetic centre at a point inside the shapes.
If a, b and c are parallel lines and they intersect with another two straight lines, r and s, then the segments between them will be proportional. ''
3 List some of the applications of Thales' theorem.
Similarity between triangles
Two similar triangles have:
• Proportional sides: ' ' ' a a b b c c ratiosimilitary == =of ratio of similitary
• Equal angles:
Two triangles in Thales’ position are similar.
Criteria for similar triangles
4 What does it mean that two triangles are in the Thales position?
5 Adapt these similarity criteria to the case of right triangles.
Similarity between right-angled triangles
Two right-angled triangles are similar if one of the acute angles of one triangle is equal to one of the acute angles of the other triangle.
• All triangles obtained by drawing lines perpendicular to one of the sides of an angle are similar.
• In a right-angled triangle, the height from the hypotenuse creates two triangles that are similar to the original.
6 Why are triangles ABC and AMB similar, and why are triangles ABC and AMC similar?
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