Дипломная работа: Методические основы подготовки будущих учителей математики в условиях полиязычного образования

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

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.

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

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 4.5.

1. If , what is d in terms of a, b, and c?

(A)

(B) c - a - b

(C) a + c - b

(D) c - ab

(E)

2. 1. If , what is c in terms of a, b, and d?

(A)

(B)

(C) a - b + d

(D) b - ad

(E) b + a - d

3. Find the solution of

(A) 245

(B) 243

(C) 255

(D) 249

(E) 241

Exercise 4.6.

1. What is the sum of the prime factors of 120?

(A) 14

(B) 12(C) 13(D) 11(E) 15

2. What is the sum of the prime factors of 150?

(A) 11

(B) 13(C) 12(D) 10(E) 14

3. What is the sum of the prime factors of 160?

(A) 11

(B) 10(C) 12(D) 14(E) 13

Exercise 4.7.

1. What is the value of