Методические основы подготовки будущих учителей математики в условиях полиязычного образования

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Ratios and Proportions

A ratio is a fraction that compares two quantities that are measured in the same units. The first quantity is the numerator, and the second quantity is the denominator.

For example, if in right , the length of leg is 6 inches and the length of leg is 8 inches, we say that the ratio of AC and BC is 6 to 8, which is often written as 6: but is just the fraction . Like any fraction, a ratio can be reduced and can be converted to a decimal or a percent.

AC to BC = 6 to 8 = 6: 8 = .

AC to BC = 3 to 4 = 3: 4 = .

If you know that AC = 6 inches and BC = 8 inches, you know that the ratio of AC to BC is 6 to 8. However, if you know that the ratio of AC to BC is 6 to 8, you cannot determine how long either side is. They may be 6 and 8 inches long but not necessarily.

Their lengths, in inches, may be 60 and 80 or 300 and 400 since and are both equivalent to the ratio . In fact, there are infinitely many possibilities for the lengths.

AC	6	3	24	2.4	300	3x
BC	8	4	32	3.2	400	4x

The important thing to observe is that the length of can be any multiple of 3 as long as the length of is the same multiple of 4.

· If two numbers are in the ratio of a: b, then for some number x, the first number is ax and the second number is bx.

· In any ratio problem, write x after each number and use some given information to solve for x.

Example.

In right triangle, the ratio of the length of the shorter leg to the length of the longer leg is 5 to 12. If the length of the hypotenuse is 65, what is the perimeter of the Triangle?

Solution.

Draw a right triangle and label it with the given information; then use the Pythagorean theorem.

So AC = 5(5) = 25, BC = 12(5) = 60, and the perimeter equals 25 + 60 + 65 = 150. Ratios can be extended to 3 or 4 or more terms. For example, we can say that the ratio of freshmen to sophomores to juniors to seniors in a school band is 3: 4: 5: 5. This means that for every 3 freshmen in the band there are 4 sophomores, 5 juniors, and 4 seniors.

Key words

mixed numbers	m?kst n?mb?s	смешанные числа
consists	k?n?s?sts	содержащий
abbreviation	?bri?v??e??n	сокращение
complex fraction	k?mpleks frжk?n	смешанная дробь
appear	??p??	появиться
mentioned previously	men?nd pri?v??sl?	упомянутый ранее
simplify	s?mpl?fa?	упростить
adding	жd??	добавление
necessary	nes?s?r?	необходимый
increase	??kri?s	увеличение
decrease	di?kri?s	уменьшение
initial quantity	??n???l kw?nt?t?	первоначальное количество
population	p?pj??le??n	население
ratio	re?????	соотношение
measured	me??d	измеренный
leg	leg	сторона
inches	?n?	дюйм
possibilities	p?s??b?l?t?s	возможности
to observe	?b?z??v	наблюдать
to solve	s?lv	решать
label	le?bl	пометить
extended	?ks?tend?d	расширенная
sophomores	s?f?m??s	второкурсник
juniors	?u?n??z	младшеклассники
seniors	si?n??z	учителя

Proportions

A proportion is an equation that states that two ratios are equivalent. Since ratios are just fractions, any equation such as , in which each side is a single fraction, is a proportion.

You can solve proportions by cross multiplying. If , then ad = bc.

Example.

If , then 20(x+3) = 19(x+5) ?20x+60=19x+95? x=35

A rate is a fraction that compares two quantities measured in different units. Rates often use the word “per” as in miles per hour and dollars per week.

· Set up rate problems just like ratio problems.

· Solve the proportions by cross multiplying.

Example.

Frank can type 600 words in 15 minutes. If Diane can type twice as fast, how many words can she type in 40 minutes?

Solution.

Since Diane types twice as fast as Frank, she can type 1200 words in 15 minutes. How handle this rate problem exactly as you would a ratio problem. Set up a proportion and cross multiply:

15x = (40)(1200) = 48000 ? x = 3200

ALGEBRA

Polynomials

A monomial is aa number or a variable or a product of numbers and variables. Each of the following is a monomial:

The numerical portion of a monomial is called the coefficient and is always written in front of the variables. The coefficient of is 7. If there is no number in front of a variable, the coefficient is 1 or -1, because x means 1x and -xy means -1xy.

A polynomial is a monomial or the sum of two or more monomials. Each monomial that makes up a polynomial is called a term of the polynomial. The terms of a polynomial are separated by addition signs (+) and subtraction signs (-). Each of the following is a monomial:



Note that and so is the sum of the monomials and - 7. Also, the sum, difference, and product of any polynomials is itself a polynomial.

The first polynomial in the preceding list ( is a monomial because it has one term. The second (, third (, fifth ( , and sixth () polynomials are called binomials because they have two terms. The fourth ( and seventh () polynomials are called trinomials because they have tree terms.

Example.

To evaluate when x = -2 and y = , rewrite the polynomial, replacing each x by -2 and each y by . Be sure to write each number in parentheses.

· The only terms of a polynomials that can be combined are like terms.

· To add two polynomials, write each in parentheses and put a plus sign between. Then erase the parentheses and combine like terms.

· To subtract two polynomials, write each one in parentheses and put a minus sign between them. Then change the minus sign to a plus sign, change the sign of every term in the second parentheses, and use previous fact to add them.

· To multiply monomials, first multiply the coefficients, and then multiply their variables (one by one) by adding their exponents.

· To multiply a polynomial by a monomial, just multiply each term of the polynomial by the monomial.

· To multiply two polynomials, multiply each term in the first polynomial by each term in the second polynomial and simplify by combining terms, if possible.

· To divide a polynomial by a monomial, divide each of term by the monomial. Then simplify each term by reducing the fractions formed by the coefficients to lowest terms and applying the laws of exponents to the variables.

· The first step in factoring a polynomial is to look for the greatest common factor of all the terms and, if there is one, to use the distributive property to remove it.

To factor a trinomial, remove a common factor, if there is one, and then use trial and error to find the two binomials whose product is that trinomial.

Example.

Algebraic fractions

Although the coefficient of any term in a polynomial can be a fraction, such as , the variable itself cannot be in the denominator. Expressions such as and , which do have variables in their denominators, are called algebraic fractions. You will have no trouble manipulating algebraic fractions if you treat them as regular fractions, using all the standard rules for adding, subtracting, multiplying, and dividing.

Whenever you have to simplify an algebraic fraction, factor the numerator and denominator, and divide out any common factors.

Example. Simplify .

Solution.



Note that is an identity. For every real number (except -2 and -3, for wgich the original fraction is undefined), the expressions have the same value. For example, when x = 8:

and

Similarly, when x = 3, both expressions equal 0, and when x = - 4, both expressions equal 7.

Equations and Inequalities

Example 1.

The following solution of the equation illustrates each of the six steps.

Step	What you Should Do
1	Get rid of fractions (and decimals) by multiplying both sides by a common denominator.	Multiply each on both sides of the equation by 2: x+6(x - 2) = 4(x + 1) + 2
2	Get rid of all parentheses by using the distributive law.	x+6x - 12 = 4x + 4 + 2
3	Combine like terms on each side.	7x - 12 = 4x + 6
4	By adding or subtracting, get all the variables on one side.	Subtract 4x from each side: 3x - 12 = 6
5	By adding or subtracting, get all the constants onto the other side.	Add 12 to each side: 3x = 18
6	Divide both sides by the coefficient of the variable.	Divide both sides by 3: x = 6



Example 2.

For what value of x is 5(x - 10) = x + 10?

Step	Question		5(x-10) = x + 10
1	Are there any fractions or decimals?	No
2	Are there any parentheses?	Yes	Distribute: 5x - 50 = x + 10
3	Are they any like terms to combine?	No
4	Are they variables on both sides?	Yes	Subtract x from each side: 4x - 50 = 10
5	Is there a constant on the same side as the variable?	Yes	Add 50 to each side: 4x = 60
6	Does the variable have a coefficient?	Yes	Divide both sides by 4: x = 15

· Memorize these six steps in order, and use this method whenever you have to solve this type of equation or inequality.

· When you have to solve for one variable in terms of others, treat all the others as if they were numbers, and apply the six-step method.

Absolute value, Radical, and fractional equations and inequalities.

Example 1. For what value of x is

Example 2. For what values of x is ?

[] ?[]?

Case 1: x > 0.

Case 2: x < 0.

Finally, by combining the two cases, we see that the solution set is {x| x < 0 or x > 16}.

Example 3. For what values of x is 3|x+5| - 5 = 7?

3|x+5| - 5 = 7 ? 3|x + 5| = 12? |x + 5| = 4 ? x + 5 = 4 or x + 5 = - 4 ? x = - 1 or x = - 9.

Quadratic Equations

A quadratic equation is an equation that can be written in the form a, where a, b, and c are any real numbers with a ? 0. Any number, x, that satisfies the equation is called a solution or a root of the equation.

Quadratic Formula

If a, b, and c are real numbers with a ? 0 and if , then

Recall that the symbol ± is read “plus or minus” and that is an abbreviation of the quadratic equation for or .

As you can see, a quadratic equation has two roots, both of which are determined by the quadratic formula.

The expression that appears under the square root symbol is called the discriminant of the quadratic equation.

Example 1. What are the roots of the equation ?

a = 1, b = -2, c = -15

and

D =

So

or

Example 2. What are the roots of the equation ?

First, rewrite the equation in the form :

Then a = 1, b = -10, c = 25

and

D =

So

or

Notice that since 10 + 0 = 10 and 10 - 0 = 10, the two roots are each equal to 5.

Example 3. Solve the equation :

a = 2, b = -4, c = -1

and

D =

So

or .

Example 4. Solve the equation :

a = 1, b = -2, c = 2

and D =

So

or

· If a, b, and c are rational numbers with a ? 0, if , and if , then

Value of Discriminant	Nature of the Roots
D = 0	2 equal rational roots
D < 0	2 unequal complex roots that are conjugates of each other
D > 0
D is a perfect square	2 unequal rational roots
D is not a perfect square	2 unequal rational roots



If , then the sum of the two roots is and the product of the two roots is .

Example. Find a quadratic equation for which the sum of the roots is 5 and the product of the roots is 5.

For simplicity, let a = 1. Then = 5?b = -5, and So the equation satisfies the given conditions.

Key words

states	ste?ts	утверждает
often	?f(t)?n	часто
monomial	m??n??m??l	одночлен
variable	ve?.ri.?.bl?	переменная
numerical portion	nju??mer.?.kl? p??.??n	числовая часть
polynomial	p?l·??no?·mi·?l	многочлен
separated	sep?re?t?d	разделенный
preceding	pr??si?.d??	предшествующий
binomials	ba??n??m??l	двучлен
trinomials	tra??n??m??l	трехчлен
replacing	r??ple?s??	заменять
combined	k?m?ba?nid	совмещенный
sign	sa?n	знак
distributive	d?s?tr?bj?t?v	распределительный
property	pr?p?ti	свойство
manipulating	m??n?p.j?.le?tin:	манипулирование
treat	tri?t	рассматривать/относиться
identity	a??dent?t?	тождество
illustrates	?l?stre?ts	показывает/поясняет
recall	r??k??l	напоминание
abbreviation	?bri?v??e??n	сокращение
notice	n??t?s	замечание
For simplicity	s?m?pl?s?t?	простота



Exponential Equations

There are two ways to handle equations of this type: use the laws of exponents or use logarithms.

There is a big difference between the equations and . The first equation is much easier to solve than the second if you recognize that 16 is a power of 2.

Example 1. For what value of x is ?

Example 2. For what value of x is ?

Since 15 is not a power of 2, you must use logarithms.

Example 3. If , what is the ratio of x to y?

Systems of linear equations

When the graphs of the equations are lines, the equations are called linear equations. A system of equations is a set of two or more equations involving two or more variables. A solution consists of a value for each variable that will simultaneously satisfy each equation.

Each of the equations 2x+y=13 and 3x-y=12 has infinitely many solutions. However, only one pair of numbers, x = 5 and y = 3, satisfies both equations simultaneously: 2(5) + 3 = 13 and 3(5) - 3 = 12. This then is the only solution of the system of equations:

There are three basic methods to solve a system of linear equations, such as the one above: two algebraic ones - the addition method and the substitution method - and one graphic one.

Addition method

Example 1.

The easiest way to solve the system of equations discussed above it to add the two equations:

Now solve for y by replacing x with 5 in either of the two original equations.

For example: 2(5) + y = 13 ? 10 + y = 13 ? y = 3

So, the unique solution is x = 5 and y = 3.

Example 2.

To solve the system , you cannot just add the two equations.

Fortunately, there is an easy remedy. If you multiply the first equation by 2, you will get an equivalent equation: 4x + 2y = 26. Now you can use the addition method to solve the new system:

Then, substitute 4 for x in one of the original equations:

2(4) + y = 13 ? 8 + y = 13 ? y = 5

So, the solution is x = 4 and y = 5.

Example 3.

To solve the system , multiply the first equation by 3 and the second equation by -2, and then add the new equations:

Now replace y by 7 in either of the original equations:

The solution is x=5 and y=7.

The substitution method

If in system of equations either variable has a coefficient of 1 or -1, solving the system by the substitution method may be just as easy or even easier than solving it by the addition method.

Now in the second equation you can replace y by 13-2x:

This is now a simple equation in one variable that you can solve using the six-step method.



Then substitute 5 for x:

The solution is x=5 and y=3.

The graphing method

System of linear equations can also be solved graphically. To solve

graph each of the lines and find the point where the two lines intersect. The x- and y-coordinates of the point of intersection are the x and y values of the solution.

To solve this problem, you can make a rough sketch.

Solving linear-quadratic systems



Example.

To solve the system , use the substitution method. Replace the y in the second equation by 2x - 1.

If x = 3, then y = 2(3) - 1 = 5; and if x = 1, then y = 2(1) - 1 = 1.

So, there are two solutions: x = 3, y = 5 and x = 1, y = 1.

Word problems

To solve word problems algebraically, you must treat algebra as a foreign language and translate “word for word” from English to algebra, just as you would from English into any foreign language. When translating into algebra, you should use some letter (often x) to represent the unknown quantity you are trying to determine. Review all of the examples in this section so that you master this translation process.

Rate problems

The basic formula used in all rate problems is

Of course, you can solve for one of the other variables:



Sometimes in rate problems, the word speed is used instead of rate. When you solve word problems, be sure you use consistent units.

Example 1.

John drove from his house to his office at an average speed of 30 miles per hour. If the trip took 40 minutes, how far, in miles, is it from John's house to his office?

Solution.

You know the rate (30mph) and the time (40 minutes), and you want to find the distance. Of course, you are going to use the formula d = rt. However, if you write d = (30)(40)=1200, you know something is wrong. Clearly, John didn't drive 1200 miles in less than one hour. The problem is that the units are wrong. The formula is really:

So, you have to convert 40 minutes to hours:

Then 60x = 40?. So it took John hours to get to his office.

Now you can see the formula d=rt, with r = 30 and t = .



hours) =

Example 2.

If Brian can paint a fence in 4 hours and Scott can paint the same fence in 6 hours, when working together, how long will it take the two of them to paint the fence?

Solution.

Call painting the fence “the job”. Then Brian works at the rate of . Similarly, Scott's rate of work is . Together they can complete ( jobs per hour. Finally,

So, it will take Brian and Scott to paint the fence.

Age problems

In age problems, it often helps to organize the given data in a table.

Example.

In 2001, Lior was four times as old as Ezra, and in 2003, Lior was twice as old as Ezra. How many years older than Ezra is Lior?

Solution.



Let x represent Ezra's age in 2001, and make the following table:

Year	Ezra	Lior
2001	x	4x
2003	x + 2	4x+2

In 2003, Lior was twice as old as Ezra, so

So in 2001, Ezra was 1 and Lior was 4. Lior is 3 years older than Ezra.

Percent problems

Example 1.

There are twice as many girls as boys in a biology class. If 30% of the girls and 45% of the boys have completed a lab, what percent of the students in the class have not yet completed the lab?

Solution.

If x represents the number of boys in the class, then 2x represents the number of girls in the class. Then of the 3x students in the class, the number of students who have completed the lab is

The fraction of students who have completed the lab is

So, 65% of the students have not yet completed the lab.



Key words

linear	l?n??	линейный
involving	?n?v?lv??	включающий в себя
consists	k?n?s?sts	состоит/заключается
simultaneously	s?ml?te?n??sl?	одновременно/вместе
substitution	s?bst??tju??n	замена
fortunately	f????n?tl?	к счастью
remedy	rem?d?	мера/средство
intersect	?nt??sekt	пересечься
foreign	f?r?n	иностранный
represent	repr??zent	представлять
rate	re?t	скорость
instead	?n?sted	вместо
consistent units	k?n?s?st?nt ju?n?t	последовательные точки/единицы
average	жv?r??	средняя
trip	tr?p	поездка/путешествие
fence	fens	забор
completed	k?m?pli?t?d	завершенный

Plane geometry

Lines and Angles

Lines are usually referred to by lowercase letters, such as l, m, and n. We can also name a line using two of the points on the line. If A and B are points on line l, we can refer to line l as a line .

represents the ray that consists of point A and all the points on

represents the line segment that consists of points A and B and all the points on that are between them.

Finally, represents the length of segment

If two line segments have the same length, we say they are congruent. The symbol is used to indicate congruence, so in the figure below we have .



Angles

An angle is formed by the intersection of two line segments, rays, or lines. The point of intersection is called the vertex of the angle.

· An acute angle measures less than .

· A right angle measures .

· An obtuse angle measures more than but less than .

An angle can be named by three points: a point on one side, the vertex, and a point on the other side, in that order. When there is no possible ambiguity, we can name the angle just by its vertex.

If two or more angles form a straight angle, the sum of their measures is

Example.

If in the figure below a:b:c=2:3:4, then a=2x, b=3x, c=4x. So,

2x+3x+4x=180 ? 9x = 180 ? x=



· The sum of the measures of all nonoverlapping angles around a point is

· Vertical angles have equal measures.

Perpendicular and parallel lines

Two lines that intersect to form right angles are called perpendicular. Two lines that never intersect are said to be parallel. Consequently, parallel lines form no angles, However, if a third line, called transversal, intersects a pair of parallel lines, eight angles are formed, and the relationship between these angles are very important.

If pair of parallel lines is cut by a transversal that is perpendicular to the parallel lines, all eight angles are right angles.

If a pair of parallel lenis is cut by a transversal that is not perpendicular to the parallel lines:

Four of the angles are acute, and four are obtuse.

All four acute angles are congruent.

All four obtuse angles are congruent.

The sum of the measures of any acute angle and any obtuse angle is



Angles d and f, as well as angles c and e, are called alternate interiors angles because they are interior to the two parallel lines and on alternate sides of the transversal.

Angles a and g, as well as angles b and h, are called alternate exterior angles because they are exterior to the parallel lines and on alternate sides of the transveral.

Pairs a and e, and f and h, and c and g are called corresponding angles because they are in corresponding positions in relationship to the parallel lines and transversal.

Pairs d and e as well as c and f are called consecutive interior angles because they are interior to the parallel lines and on the same of the transversal.

Triangles

Sides and angles of a triangle

In any triangle, the sum of the measures of the three angles is



The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.

Also, in any triangle:

· The longest side is opposite the largest angle.

· The shortest side is opposite the smallest angle.

· Sides with equal lengths are opposite angles with equal measures (the angles opposite congruent sides are congruent).

· A triangle is called scalene if the three sides all have different lengths. Then by previous fact, the three angles all have different measures.

· A triangle called isosceles if two sides are congruent.

· A triangle is called equilateral if all three sides are congruent. Since the sum of the measures of three angles is , each angle is

· Acute triangles are triangles in which all three angles are acute. An acute triangle could be scalene, isosceles, or equilateral.

· Obtuse triangles are triangles in which one angle is obtuse and two are acute. An obtuse triangle could be scalene or isosceles.

· Right triangles are triangles that have one right angle and two acute angles. A right triangle could be scalene or isosceles. The side opposite the angle is called the hypotenuse, it is the longest side. The other two sides are called the legs.



Right triangles

If a and b are the measures in degrees, of the acute angles of a right triangle, 90 + a + b = 180 ? a + b = 90.

In any right triangle, the sum of measures of the two acute angles is

Example.

To find the average of a and b in below, a + b = 90, so

Pythagorean theorems and corollaries

Let a, b and c be the lengths of the sides of , with a ? b ? c.

· if and only if angle C is a right angle.

· if and only if angle C is obtuse.

· if and only if angle C is acute.

For any positive number x, there is a right triangle whose sides are 3x, 4x, 5x.

For example:

x = 13, 4, 5x = 515, 20, 25

x = 26, 8, 10x = 1030, 40, 50

x = 39, 12, 15x = 50150, 200, 250

x = 412, 16, 20x = 100300, 400, 500

Other right triangles with integer sides that you should recognize immediately are the ones whose sides are 5, 12, 13 and 8, 15, 17. These sets of three numbers are often referred to as Pythagorean triples.

Special right triangles

Let x be the length of each leg and let h be the length of the hypotenuse of an isosceles right triangle. By the Pythagorean theorem,

In 45-45-90 right triangle, the sides are x, x, and .

· If you are given the length of a leg, multiply it by to get the length of the hypotenuse.

· If you are given the length of the hypotenuse, divide it by to get the length of each leg.



h/

In 30-60-90 right triangle, the sides are x, and 2x.

If you know the length of the shorter leg (x):

· Multiply it by to get the length of the longer leg.

· Multiply it by 2 go get the length of the hypotenuse.

If you know the length of the longer leg (a):

· Divide it by to get the length of the shorter leg.

· Multiply the length of the shorter leg by 2 to get the length of the hypotenuse.

If you know the length of the hypotenuse (h):

· Divide it by 2 to get the length if the shorter leg.

· Multiply the length of the shorter leg by to get the length of the longer leg.

Perimeter and area

The perimeter of a triangle is the sum of the lengths of the three sides.

Example.

To find the perimeter of an equilateral triangle whose height is 12, note that the height divides the triangle into two 30-60-90 right triangles.



AD = and s =; s = 2

So the perimeter is 3s = 24.

Triangle inequality

· The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

· The difference of the lengths of any two sides of a triangle is less than the length of the third side.

The area of a triangle is given by A = , where b and b are the length of the base and height, respectively.

(1) Any side of the triangle can be taken as the base.

(2) The height (which is also called altitude) is a line segment drawn perpendicular to the base from the opposite vertex.

(3) In a right triangle, either leg can be the base and the other the height.

(4) If one endpoint of the base is the vertex of an obtuse angle, then the height will be outside the triangle.

If A represents the area of an equilateral triangle with side s, then A=.

If a, b, and c are the lengths of the three sides of a triangle, and if s represents the semiperimeter, , then the area of the triangle is given by A =

Similar triangles

Two triangles, such as triangle I and triangle II in the future below, that have the same shape but not necessarily the same size are said to be a similar.

Two triangles are similar provided the following two conditions are satisfied:

The lengths of the corresponding sides of the two triangles are in proportion.

If the measures of two angles of one triangle equal to the measures of two angles of a second triangle, then the triangle are similar.

A line that intersects two sides of a triangle and is parallel to the third side creates a smaller triangle that is similar to the original one.

If two triangles are similar, and if k is the ratio of similitude:

· The ratio of all their linear measurements is k.

· The ratio of their areas is



Key words

Lines	la?ns	прямая
referred	r??f??d	относится
lowercase	l???(r)ke?s	строчная буква
ray	re?	луч
line segment	la?n ?segm?nt	отрезок прямой
length	le?и	длина
congruent	k??gr??nt	соответствующий
angle	Ж?gl	угол
vertex	v??teks	вершина
acute	??kju?t	острый
measures	me??s	измерения
obtuse	?b?tju?s	тупой
ambiguity	жmb?'gju??t?	Многозначность/неоднозначность
consequently	k?ns?kw?ntl?	следовательно
transversal	trжnz'v??s?l	поперечный
relationship	r??le??n??p	связь
alternate interiors angles	?n?t??r??s	внутренние углы
alternate exterior angles	?k?st?(?)r??	внешние углы
corresponding angles	k?r?s?p?nd??	соответствубщие углы
consecutive interior angles	k?n?sekj?t?v	последовательные внутренние углы
opposite	?p?z?t	противоположный
scalene	ske?li?n	разносторонний
isosceles	a??s?s?li?z	равнобедренный
equilateral	i?kw?'lжt(?)r?l	равносторонний
below	b??l??	ниже
corollaries	k??r?l?r?	последствия/заключения
respectively	r?s?pekt?vl?	соответственно
semiperimeter		полупериметр
provided	pr??va?d?d	предусмотренный/представленный
similitude	s??m?l?tju?d	подобие

Quadrilaterals and Other polygons

A polygon is a closed geometric figure made up of line segments. The line segments are called sides, and the endpoints of the line segments are called vertices (each one is called a vertex). Line segments iside the polygon drawn from one vertex to another are called diagonals.

Three-sided polygons, called triangles. Although in this section our main focus will be on four-sided polygons, which are called quadrilaterals, we will discuss other polygons as well. There are special names for many polygons with more than four sides.

Number of sides	Name	Number of sides	Name
5	Pentagon	8	Octagon
6	Hexagon	7	Decagon

A regular polygon is a polygon in which al the sides have the same length and all the angles have the same measure. A regular three-sided polygon is an equilateral triangle, and, as we shall see, a regular quadrilateral is a square.

The angels of a polygon

A diagonal of a quadrilateral divides it into two triangles. Since the sum of the measures of the three angles in each of the triangles , the sum of the measures of the angles in the quadrilateral is

In any quadrilateral, the sum of the measures of the four angles is

Similarly, any polygon can be divided into triangles by drawing in all of the diagonals emanating from one vertex.

The sum of the measures of the n angles in a polygon with n sides is (n-2)Ч.

Example 1.

To find the measures of each angle of regular octagon, first use previous fact to get that the sum of all eight angles is (8-2)Ч=6Ч. Then since in a regular octagon all eight angles have the same measure, the measure of each one is ч8 = .

An exterior angle of a polygon is formed by extending a side. Surprisingly, in all polygons, the sum of the measures of the exterior angles is the same.

In any polygon, the sim of the measures of the exterior angles, taking one at each vertex, is .

Example 2.

Previous fact gives us an alternative method of calculating in the measure of each angle in a regular polygon. The sum of the measures if the eight exterior angles of any octagon is . As a result, in a regular octagon, the measure of each exterior angle is . Therefore, the measure of each interior angle is .



Special quadrilaterals

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Any side of a parallelogram can be its base, and a line segment drawn from a vertex perpendicular to the opposite base is called the height.

Parallelograms have the following properties illustrated in the figures below:

· Opposite sides are parallel: and

· Opposite sides are congruent: and

· Opposite angles are congruent: and .

· The sum of the measures of any pair of consecutive angles is .

· A diagonal divides the parallelogram into two congruent triangles

· The two diagonals bisect each other: AE=EC and BE=ED.

A rectangle is a parallelogram in which all four angles are right angles.

Since a rectangle is a parallelogram, all of the properties listed in previous fact hold for rectangles. In addition:

· The measure of each angle in a rectangle is

· The diagonals of a rectangle have the same length: AC = BD.

A rhombus is a parallelogram in which all four sides have the same length.

Since a rhombus is a parallelogram, all of the properties of parallelograms hold for rhombuses too. In addition:

· The length of each side of a rhombus is the same.

· The two diagonals of a rhombus are perpendicular.

· The diagonals of a rhombus are angle bisectors.

A square is a rectangle in which all four sides have the same length. So a square is both a rectangle and a rhombus.

Since a square is a rectangle and a rhombus, all of the properties listed in previous key facts hold for squares.

A trapezoid is a quadrilateral in which exactly one pair of opposite sides a parallel. The parallel sides are called the base of trapezoid, and the distance between the two bases is called the height. If the two nonparallel sides are congruent, the trapezoid is called isosceles and, in that case only, the diagonals are congruent.

Perimeter and area of quadrilaterals

The perimeter (P) of a polygon is the sum of the lengths of all of its sides. The area (A) of a polygon is the amount of space it enclosed.

For a rectangle: P = 2(l+w)

For a square: P = 4s



Areas

· For a parallelogram: A = bh

· For a rectangle: A = lw

· For a square: A = or A =

· For a trapezoid: A =

Example 1.

What are the perimeter and area of a rhombus whose diagonals are 6 and 8? First draw and label a rhombus.

Since the diagonals bisect each other, BE = ED = 3 and AE = EC = 4. Also, since the diagonals of a rhombus are perpendicular, ?BEA is a right angle and is a 3-4-5 right triangle. So AB = 5 and the perimeter of the rhombus is 4Ч5=20. The easiest way to calculate the area of rhombus is to recognize that it is the sum of the areas of four 3-4-5 right triangles. Since each triangle has an area of , the area of rhombus is 4Ч6=24.

Example 2.

In the figure below, the area of parallelogram ABCD is 40 What are the areas of rectangle AFCE, trapezoid AFCD, and triangle BCF?

Since the base of parallelogram ABCD is 10 and its area is 40, its height, AE, must be 3. Then must be a 3-4-5 right triangle with DE = 3, which implies that EC = 7. So the area of rectangle AFCE is 7Ч4 = 28; the area of trapezoid AFCD is and the area of each small triangle is

Circles

A circle consists of all the points that are the same distance from one fixed point called the center. That distance is called the radius of the circle. The figure below is a circle of radius 1 unit whose center is at the point. O, A, B, C, D, and E, which are each 1 unit from O, are all points on circle O. The word radius is also used to represent any of the line segments joining the center and a point on the circle. The plural of radius is radii. In circle O below and are all radii. If a circle has radius r, each of the radii is r units long. A point is inside a circle if the distance from the center to that point is less than the radius. A point is outside a circle if the distance from the center to that point is greater than the radius.



Example 2.

In the figure below, O is the center of the circle. To find m?B and m?O, observe that since and are radii, OA = OB and is isosceles. So m?B = and m?O = .

Any triangle formed by connecting the endpoints of two radii is isosceles

A chord of a circle is a line segment that has both endpoints on the circle. In the figure at the beginning of this chapter, and are chords. A chord such as that passes through the center is called a diameter. Since BE = EO + OB, a diameter is twice as long as a radius.

· If d is the diameter and r is the radius of a circle: d = 2r.

· Diameters are the longest line segments that can be drawn that have both endpoints on or inside a circle.

Circumference and area

The total length around a circle is called the circumference. In every circle, the ratio of the circumference to the diameter is exactly the same and is denoted by the symbol р.

So, there are two formulas for the circumference of a circle:

The value of р is approximately 3.14.

Example 1. If the circumference of a circle is equal to the perimeter of a square whose sides are 12, what is the radius of the circle?

Solution.

Since the perimeter of the square is 4Ч12 = 48:

2рr = 48 ?

An arc consists of two points on a circle and all the points between them. If two points, such A and B in circle O, are the endpoints of a diameter, they divide the circle into two arcs called semicircles. In we wanted to refer to the larger arc going from X to Y, the one through A and B, we would refer to it as arc or arc .

The degree measure of a circle is .

An angle such as ?AOB in the figure below, whose vertex is at the center of a circle, is called a central angle.

The degree measure of an arc equals the degree measure of the central angle that intercepts it.

In the figure above, how long is arc ? Since the radius of circle P is 12, its diameter is 24 and its circumference is 24р. Since there are in a circle, arc , or , of the circumference: .

The formula for the area of a circle of radius r is A =

In a circle of radius r, if an arc measures

· The length of the arc is

· The area of the sector formed by the arc and two radii is .

Example.

What are the perimeter and area of the shaded region in the figure below?



The circumference of the circle is Since arc is of the circle, the length of arc is the hypotenuse of isosceles right triangle POQ, PQ = 10. So the perimeter of the shaded region is 10 + 5р. Since the area of the circle is р=р(, the area of sector POQ is The area of POQ = So the area of the shaded region is 25р - 50.

An angle formed by two chords with a common endpoint is called an inscribed angle. In the figure below, ?ABC, ?ADC, ?BAD, and ?BCD are all inscribed angles.

The measure of an inscribed angle is one-half the measure of its intercepted arc.

Example.

To find m?ABC in circle O in the figure below, observe that since the measure of an arc is equal to the measure of the central angle that intercepts it, the measure of is Since ?ABC is an inscribed angle, its measure is one-half the measure of arc

Tangents to a circle

A line or line segment to a circle if it intersects the circle exactly once.

Line l is tangent to both circles.



and are each tangent to the circle.

· From any point outside a circle, exactly two tangets can be drawn to the circle.

· If two tangents are drawn from a point P outside a circle, intersecting the circle at A and B, then PA=PB.

· The measure of the angle formed by two tangents drawn from the same point is one-half the difference of the two intercepted arcs.

· A line tangent to a circle is perpendicular to the radius (or diameter) drawn to the point of contact.

· When a square is inscribed in a circle, the diagonals of the square are diameters of the circle. ( is a diagonal and a diameter).

· When a circle is inscribed in a square, the length of a diameter is equal to the length of side of the square. (AB=WX)

Key words

polygon	p?l???n	многоугольник
vertices	v??teks	вершина
quadrilaterals	kw?dr?'lжt?r?l	четырехугольник
therefore	рe?f??	следовательно
properties	pr?p?t?s	свойства
bisect	ba??sekt	раздваивать/делить попалам
rhombus	r?mb?s	ромб
bisector	ba??sekt?	биссектриса
listed	l?st?d	перечисленный
pair	pe?s	пара
in that case	?n ржt ke?s	в таком случае
enclosed	?n?kl??zd	закрытый
joining	???n??	присоединяться/объединяться
plural	pl??r?l	множество
chapter	?жpt?	глава/раздел
passes through	p??sis иru?	проходить/через
circumference	s??k?mf?r?ns	длина окружности
approximately	??pr?ks?m?tl?	приближенно
arc	??k	дуга
shaded	?e?d?d	затемненный
inscribed	?n?skra?bd	вписанный

Следующие задания предназначены для самостоятельного решения и оценки знаний уровня технического английского языка.

Слова, подчеркнутые и выделенные красным цветом или жирным шрифтом следует рассматривать как обязательные для заучивания и являются ключевыми в задании, исходя из своей значимости с точки зрения критического мышления и изучения иностранного языка.

Подчеркнутые слова - являются ключевыми в задании и помогут быстро найти верный курс к решению задания.

Например:

1. If and 5y = 20, what is the value of x - y?

Словосочетания «What is» и термин «value» дают полное представление о смысле задания, в котором необходимо определить значение выражения « х - y ». Или:

2. What is the area of an equilateral triangle whose altitude is 6?

В данном примере необходимо определить площадь прямоугольного треугольника, зная его высоту.

3. ALGEBRA (Check yourself)

Exercise 4.1

1. || - 2| - | - 3| - | - 6|| =

(A) - 2

(B) 2

(C) -7

(D) -5

(E) 7

2. |9 - | - 4| - | - 6|| =

3. (A) - 3

4. (B) 2

5. (C) - 1

6. (D) 1

7. (E) 3

3. || - 7| - | - 9| - | - 3|| =

8. (A) - 4

9. (B) 6

10. (C) - 1

11. (D) 4

12. (E) 5

4. || - 4| - | - 1| - | - 7|| =

13. (A) - 4

14. (B) 3

15. (C) - 3

16. (D) 4

17. (E) 2

Exercise 4.2.

1. What is the sum of the product and quotient of - 4 and 4?

(A) 17

(B) - 17

(C) 4

(D) 2

(E) - 4

2. What is the product of -6 and 3?

(A) 18

(B) - 18

(C) 9

(D) 3

(E) - 9

3. What is the sum of the product and quotient of - 9 and 9?

(A) - 1

(B) 1

(C) - 82

(D) 82

(E) - 9

4. What is the quotient of - 7 and 7?

(A) - 2

(B) 2

(C) - 1

(D) 1

(E) 0

Exercise 4.3.

Which of the following statements are true?

1. The product of the integers from - 5 to 4 is equal to the product of the integers from -4 to 5.

2. The sum of the integers from - 5 to 4 is equal to the sum of the integers from - 4 to 5.

3. The absolute value of the sum of the integers from - 4 to 5 is equal to the sum of the absolute values of the integers from - 4 to 5.

(A) 1 only

(B) 3 only

(C) 1 and 3 only

(D) 1 and 2 only

(E) 1, 2, and 3

Exercise 4.4.

1. When the positive integer n is divided by 5, the remainder is 3. What is the remainder when is divided by 5?

(A) 4

(B) 3

(C) 2

(D) 5

(E) 6

2. When the positive integer n is divided by 9, the remainder is 2. What is the remainder when is divided by 3?

(F) 4

(G) 3

(H) 2

(I) 1

(J) 5

3. When the positive integer n is divided by 9, the remainder is 2. What is the remainder when is divided by 3?

(K) 4

(L) 3

(M) 2

(N) 1

(O) It cannot be determined from the information given.

Exercise 4....