digital text book

Download Digital Text Book

Post on 03-Feb-2016

17 views

Category:

Documents

0 download

Embed Size (px)

DESCRIPTION

Digital Text Book

TRANSCRIPT

  • MATHEMATICS DIGITAL

    TEXT BOOK

    CLASS IX

    BINCY S BABY

    B. Ed Mathematics

    Reg. No : 18014350005

  • CONTENT

    Chapter 1

    Simple triangles

    Of the angles and sides1

    Of the sides and angles3

    Similarity.6

    Third way.7

    Chapter 2 Polynomials

    Polynomial..8

    Polynomial peculiarities9

    Addition and subtraction.10

    Polynomial multiplication.11

    Multiplication and addition........11

    Degree of a polynomial..............12

    Meaning of remainder................13

    Doing Division............................13

  • Chapter 1

    SIMILAR TRIANGLES

    OF ANGLES AND SIDES

    Here a, b, c are the lengths of the sides of

    a triangle and x, y, z are the lengths of the sides

    of another triangle with the same angles.

    We started with the triangles ABC and

    XYZ in which A=X, B=Y, C=Z.

    The numbers pairs (x, a), (y, b), (z, c) are

    the lengths of the sides opposite the pairs of

    equal angles.

    Z Y

    X

    y

    x

    z b

    a

    c

    C B

    A

    1

  • Now

    is the number which shows how

    many times the number x is the number a (or

    what part of x is a).The other numbers

    and

    have similar meanings.

    Whatever times (or part) we take to change

    the side of length a to the side of length x,

    such times(or part) is to be taken to change the

    side of length b to the side of length y and the

    side of length c to the side of length Z.

    In other words, all these pairs of lengths

    are in the same ratio. So, we can state this result

    about the angles lengths of the sides of a

    triangle with the same angles, as follows:

    If all the angles of a triangle are equal to

    the angles of another triangle, then all the pairs

    of sides opposite equal angles have the same

    ratio.

    2

  • OF SIDES AND ANGLES

    Does the pair of triangles with proportional

    sides also have the same angles?

    Lets consider the triangle shown below

    We have

    =

    =

    We want to check whether the triangles

    have the same angles.

    For that, what we do is draw a third triangle

    with the same angles of ABC, and which is

    congruent to XYZ.

    b

    a

    c

    C B

    A

    y

    x

    z

    Z Y

    X

    3

  • In PQR,

    OR = x, Q=B, R=C

    So, we must also have P=A. Thus the angles

    of PQR are equal to the angle of ABC.

    So, by the theorem we have proved,

    =

    =

    We actually draw PQR with QR = x, So,

    b

    a

    c

    C B

    A

    y

    x

    z

    R Q

    P

    y

    x

    z

    Z Y

    X

    4

  • =

    =

    We started with

    =

    =

    Comparing the last two questions, we get

    =

    and

    =

    From these, we get

    RP = y and PQ = z

    b

    a

    c

    C B

    A

    y

    x

    z

    R Q

    P

    y

    x

    z

    Z Y

    X

    5

  • Thus the three sides of PQR are equal to the

    three sides of XYZ and so these triangles are

    congruent. So, the angles opposite their pairs of

    equal sides are also equal:

    X=P, Y=Q, Z=R

    We have already seen that

    P=A, Q=B, R=C

    Thus we have

    X=A, Y=B, Z=C

    How do we state the result just proved?

    If the sides of a triangle are proportional to the

    sides of another triangle, then the angles

    opposite such sides are equal

    SIMILARITY

    If the angles of a triangle are all equal to the

    angles of another triangle, then the sides of the

    6

  • two angles are proportional: and on the

    otherhand, if the lengths of the sides of another

    triangle, then the angles of one triangle are equal

    to the angles of the other.

    Two triangles related in this way are said to

    be similar.

    THIRD WAY

    We have seen two ways to show that two

    triangles are similar: either proves that they have

    the same angles or prove that their sides are

    proportional.

    There is a third way; if a pair of sides is

    proportional and their included angles are equal,

    then also the triangles are similar.

    if two sides of a triangle are proportional

    to two sides of another triangle and if their

    included angles are equal, then the triangles are

    similar.

    7

  • Chapter 2

    POLYNOMIALS

    POLYNOMIALS

    Various positive integral powers of x are

    multiplied by specific numbers (if may be by 1

    also), the products are added or subtracted and

    finally a specific number is added or subtracted

    (it may be 0) such algebraic expression are

    called polynomials.

    A polynomial is usually written with the

    exponents of x in descending order, the number

    without x at the end.

    Thus for example, 35x-24x2+4x3 is usually

    written as 4x3-24x2+35x; and 15+3x as 3x+15.

    8

  • POLYNOMIAL PECULARITIES

    Look at 2x3-3x2+x-5. We can write it as

    2x3+(-3)x2+x+(-5)

    Each such addend is called a term of the

    polynomial.

    Thus in the polynomial we considered just

    now, the terms are 2x3, -3x2, x, -5.

    Now lets look at the terms themselves.

    Apart from the last term, all others are got by

    multiplying (Positive integral) powers of x by

    specific numbers. Each such multiplier is called

    the Co-efficient of the corresponding power.

    Thus in our example, the coefficient of x3

    is 2, the coefficient of x2 is -3 and the coefficient

    of x is 1.

    Thus the number added at the end is called

    the constant term.

    9

  • ADDITION AND SUBTRACTION

    Lets simplify (2x+3) + (4x+6)

    (2x+3) + (4x+6) = 2x+3 + 4x+6

    = (2x+ 4x) + (3 +6)

    = 6x+9

    (2x +3) - (4x+6) = 2x+3 - 4x-6

    = (2x- 4x)+(3 -6)

    = - 2x-3

    (4x2+3x+1)+(2x2+5x-2)

    = (4+2) x2+ (3+5) x+(1-2)

    = 6x2+8x-1

    (4x2+3x+1) - (2x2+5x-2)

    = (4 - 2)x2+(3 - 5)x+(1 + 2)

    = 2x2 - 2x + 3

    10

  • POLYNOMIAL MULTIPLICATION

    (x+y) (u+v) = xu+xv+yu+yv

    ie, Each number in the first sum should be

    multiplied by each number in the second sum

    and all these products should be added.

    (2x+5)(4x+3

    = (2x4x) + (2x 3) + (54x) + (53)

    = 8x2+6x+20x+15

    = 8x2+26x+15

    MULTIPLICATION AND ADDITION

    We can see that for any 3 polynomials

    p(x), q(x), r(x).

    p(x)r(x)+q(x)r(x) = (p(x)+q(x))r(x)

    11

  • Example:

    (2x+3)(x+1)+( x+4)(x+1)

    =((2x+3)+( x+4)) (x+1)

    = (3x+7) (x+1)

    = 3x2+10x+7

    DEGREE OF POLYNOMIAL

    The polynomial P(x) =6x4+4x3+5x+1.The

    power of x in it are x4, x3, x. The largest

    exponent among them is 4. It is called the

    degree of the polynomial.

    A polynomial whose degree is 1 is called a

    first degree polynomial. A polynomial whose

    degree is 2 is called a second degree polynomial

    and so on.

    12

  • MEANING OF REMAINDER

    In general, when a polynomial cannot be

    completely divided by another polynomial, the

    remainder should be a polynomial of degree less

    than that of the divisor, or a number.

    DOING DIVISION

    We want to find the quotient and reminder

    when 4x2+5 is divided by 2x+3.As taking the

    quotient as ax+b and the remainder as c, we get

    here

    (ax+b)(2x+3)+c = 4x2+5

    ie, 2ax2+(3a+2b)x+(3b+c) = 4x2+5

    Therefore, 2a= 4, 3a+2b=0, 3b+c=5

    ie, 2a=4 ; a=2

    For a=2 and 3a+2b=0.We must have 2b= -3a =-6

    ie, b= -3 and for b= -3 and 3b+c=5,

    Therefore, c=5-3b=14

    Thus the quotient is (2x-3) and the remainder 14

    13