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Transcript
IX_Maths Chapter Notes 1 IX Mathematics Chapter 1: Number Systems Chapter Notes Key Concepts 1. Numbers 1, 2, 3……., which are used for counting are called Natural numbers and are denoted by N. 2. 0 when included with the natural numbers form a new set of numbers called Whole number denoted by W 3. -1,-2,-3……………..- are the negative of natural numbers. 4. The negative of natural numbers, 0 and the natural number together constitutes integers denoted by Z. 5. The numbers which can be represented in the form of p/q where q 0 and p and q are integers are called Rational numbers. Rational numbers are denoted by Q. If p and q are coprime then the rational number is in its simplest form. 6. Irrational numbers are the numbers which are non-terminating and non-repeating. 7. Rational and irrational numbers together constitute Real numbers and it is denoted by R. 8. Equivalent rational numbers (or fractions) have same (equal) values when written in the simplest form. 9. Terminating fractions are the fractions which leaves remainder 0 on division. 10. Recurring fractions are the fractions which never leave a remainder 0 on division. 11. There are infinitely many rational numbers between any two rational numbers. 12. If Prime factors of the denominator are 2 or 5 or both only. Then the number is terminating else repeating/recurring. 2 13. Two numbers p & q are said to be co-prime if, numbers p & q have no common factors other than 1. 14. The decimal expansion of rational number is either terminating or non-terminating recurring 15. The decimal expansion of an irrational number is non-terminating, non-recurring. 16. Real numbers satisfy the commutative, associate and distributive law of addition and multiplication. 17. Commutative law of addition: If a and b are two real numbers then, a+b=b+a 19. Commutative law of multiplication: If a and b are two real numbers then, a. b = b. a 20. Associative law of addition: If a, b and c are real numbers then, a + (b + c) = (a + b) + c 21. Associative law of multiplication: If a, b and c are real numbers then, a. (b. c) = (a. b). c 22. Distributive of multiplication with respect to addition: If a, b and c are real numbers then, a. (b+ c) = a. b + a. c 23. Removing the radical sign from the denominator is called rationalisation of denominator. 24. The multiplication factor used for rationalising the denominator is called the rationalising factor. 25. The exponent is the number of times the base is multiplied by itself. 26. In the exponential representation am , a is called the base and m is called the exponent or power. 3 27. If a number is to the left of the number on the number line, it is less than the other number. If it is to the right, then it is greater than the number. 28. There is one to one correspondence between the set of real numbers and the set of point on the number line. 30. Irrational numbers like 2, 3, 5… n , for any positive integer n can be represented on number line by using Pythagoras theorem. 31. The process of visualisation of representation of numbers on the number line through a magnifying glass is known as the process of successive magnification. Key Formulae: 1. Rational number between two numbers x and y = 2. xy 2 Irrational number between two numbers x and y xy, if x andybothareirrationalnumbers xy, if xisrationalnumber andyisirrationalnumber xy, if x yisnot aperfect square andx,ybotharerationalnumbers 3. Irrational number between two rational number x and y = xy , if and only if x y is not a perfect square. 4. Irrational number between a rational number x and irrational number y= xy 5. 2 xy x = y2 6. 3 xy x = y3 7. n xy x = yn 4 8. Square root identities: For real numbers a>0 and b>0 ab a a b b ( a ( a (a a b ( b)( b)( c b)(a a a b) a b d) ac b) a2 b b)2 a b 2 bc ad bd ab 1 n n a a an where a >0 and is a real number and n is positive integer. 9. m m 10. a n n a n am where a, n > 0 and ‘a’ is a real number, m and n co prime integers. 11. a0 1 , where a is a real number. 12. Pythagoras Theorem: (AB) 2 + (BC) 2 = (AC) 2 where AC is hypotenuse, AB and BC are the sides of the right triangle. 13. x x 1 2 x 1 2 2 2 1 IX Math Ch 2: Polynomials Chapter Notes Top Definitions 1. A polynomial p(x) in one variable x is an algebraic expression in x of the form p(x) = a xn a xn1 a xn2 ........ a x 2 a x a , where n n1 n2 2 1 0 (i) a0 ,a1 ,a2 ......an are constants (ii) x 0 ,x1 ,x 2 ......xn are variables (iii) a0 ,a1 ,a2 ......an are respectively the coefficients of x n n1 (iv) Each of an x an 1x ,an called a term of a polynomial. n2 2x ,........a2 x 2 0 ,x1 ,x 2 ......xn . ,a1x,a0 , with an 0, is 2. A leading term is the term of highest degree. 3. Degree of a polynomial is the degree of the leading term. 4. A polynomial with one term is called a monomial. 5. A polynomial with two terms is called a binomial. 6. A polynomial with three terms is called a trinomial. 7. A polynomial of degree 1 is called a linear polynomial. It is of the form ax+b. For example: x-2, 4y+89, 3x-z. 8. A polynomial of degree 2 is called a quadratic polynomial. It is of the form ax2 + bx + c. where a, b, c are real numbers and a 0 For example: x 2 2x 5 etc. 9. A polynomial of degree 3 is called a cubic polynomial and has the general form ax3 + bx2 + cx +d. For example: x 3 2x 2 2x 5 etc. 10. A bi-quadratic polynomial p(x) is a polynomial of degree 4 which can be reduced to quadratic polynomial in the variable z = x2 by substitution. 2 11. The zero polynomial is a polynomial in which the coefficients of all the terms of the variable are zero. 12. Remainder theorem: Let p(x) be any polynomial of degree greater than or equal to one and let a be any real number. If p(x) is divided by the linear polynomial x – a, then remainder is p(a). 13. Factor Theorem: If p(x) is a polynomial of degree n≥ 1and a is any real number then (x-a) is a factor of p(x), if p(a) =0. 14. Converse of Factor Theorem: If p(x) is a polynomial of degree n≥ 1and a is any real number then p(a) =0 if (x-a) is a factor of p(x). 15. An algebraic identity is an algebraic equation which is true for all values of the variables occurring in it. Top Concepts 1. The degree of non-zero constant polynomial is zero. 2. A real number ‘a’ is a zero/ root of a polynomial p(x) if p (a) = 0. 3. The number of real zeroes of a polynomial is less than or equal to the degree of polynomial. 4. Degree of zero polynomial is not defined. 5. A non zero constant polynomial has no zero. 6. Every real number is a zero of a zero polynomial. 7. Division algorithm: If p(x) and g(x) are the two polynomials such that degree of p(x) degree of g(x) and g(x)≠ 0, then we can find polynomials q(x) and r(x) such that: p (x) = g(x) q(x) + r(x) where, r(x) =0 or degree of r(x) < degree of g(x). 8. If the polynomial p(x) is divided by (x+a), the remainder is given by the value of p (-a). 9. If the polynomial p(x) is divided by (x-a), the remainder is given by the value of p (a). 3 10. If p (x) is divided by ax + b = 0; a 0, the remainder is given by p b ; a 0. a 11. If p (x) is divided by ax - b = 0 , a 0 , the remainder is given by p b ; a 0. a 12. A quadratic polynomial ax2 + bx+ c is factorised by splitting the middle term bx as px +qx so that pq =ac. 13. The quadratic polynomial ax2 + bx+ c will have real roots if and only if b2-4ac ≥ 0. 14. For applying factor theorem the divisor should be either a linear polynomial of the form x-a or it should be reducible to a linear polynomial. Top Formulae 1. Quadratic identities: a. b. c. d. x y 2 x2 2xy y2 x y 2 x2 2xy y2 x y (x y) x 2 y2 x a (x b) x 2 (a b)x ab x y z 2 x2 y2 z2 2xy 2yz 2zx e. Here x, y, z are variables and a, b are constants 2. Cubic identities: a. b. x y 3 x y 3 x3 y3 3xy(x y) x3 y3 3xy(x y) c. x3 y3 (x y)(x2 xy y2 ) d. x3 y3 (x y)(x2 xy y2 ) e. x3 y3 z3 3xyz (x y z)(x2 y2 z2 xy yz zx) f. If x y z 0 then x3 y3 z3 3xyz Here, x, y & z are variables. 1 IX Mathematics Chapter 3: Coordinate Geometry Points to Remember Key Concepts 1. Two perpendicular number lines intersecting at point zero are called coordinate axes. The horizontal number line is the x-axis (denoted by X’OX) and the vertical one is the y-axis (denoted by Y’OY). 2. The point of intersection of x axis and y axis is called origin and denoted by ‘O’. 3. Cartesian plane is a plane obtained by putting the coordinate axes perpendicular to each other in the plane. It is also called coordinate plane or xy plane. 4. The x-coordinate of a point is its perpendicular distance from y axis. 5. The y-coordinate of a point is its perpendicular distance from x axis. 6. The point where the x axis and the y axis intersect is represented by coordinate points (0, 0) and is called the origin. It is denoted by ‘O’ on a Cartesian plane. 7. The abscissa of a point is the x-coordinate of the point. 8. The ordinate of a point is the y-coordinate of the point. 9. If the abscissa of a point is x and the ordinate of the point is y, then (x, y) are called the coordinates of the point. 10. The axes divide the Cartesian plane into four parts called the quadrants (one fourth part), numbered I, II, III and IV anticlockwise from OX. 11. The origin O has zero distance from both the axes. 12. The coordinate of a point on the x axis are of the form (x,0) and that of the point on y axis are (0,y) 13. Sign of coordinates depicts the quadrant in which it lies. The coordinates of a point are of the form (+, +) in the first quadrant, (-, +) in the second quadrant, (-,-) in the third quadrant and (+,-) in the fourth quadrant. 2 14. To plot a point P (3, 4) in the Cartesian plane. Start from origin count 3 units on the positive x axis then move 4 units towards positive y axis and mark the point P. 15. If x ≠ y, then (x,y)≠(y,x) and if (x,y) = (y,x), then x=y. 1 IX Mathematics Chapter 4: Linear Equations in Two Variables Chapter Notes Top Definitions 1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. 2. A linear equation in two variables is represented geometrically by a straight line the points of which make up the collection of solutions of equation. This is called the graph of the linear equation. Top Concepts 1. A linear equation in two variables has infinitely many solutions. 2. The graph of every linear equation in two variables is a straight line. 3. x = 0 is the equation of the y – axis and y = 0 is the equation of the x–axis. 4. The graph of x = k is a straight line parallel to the y –axis. 5. The graph of y = k is a straight line parallel to the x – axis. 6. An equation of the type y = mx represents a line passing through the origin, where m is a real number. 7. Every point on the line satisfies the equation of the line and every solution of the equation is a point on the line. 8. The solution of a linear equation is not effected when: (i) The same number is added or subtracted from both the side of an equation. (ii) Multiplying or dividing both the sides of the equation by the same non zero number. 2 Top Diagrams 1. Graph of a line passing through the origin. 2. Graph of a line parallel to x axis. 3 3. Graph of a line parallel to y axis. 1 IX Mathematics Chapter 4: Linear Equations in Two Variables Chapter Notes Top Definitions 1. An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that a and b are not both zero, is called a linear equation in two variables. 2. A linear equation in two variables is represented geometrically by a straight line the points of which make up the collection of solutions of equation. This is called the graph of the linear equation. Top Concepts 1. A linear equation in two variables has infinitely many solutions. 2. The graph of every linear equation in two variables is a straight line. 3. x = 0 is the equation of the y – axis and y = 0 is the equation of the x–axis. 4. The graph of x = k is a straight line parallel to the y –axis. 5. The graph of y = k is a straight line parallel to the x – axis. 6. An equation of the type y = mx represents a line passing through the origin, where m is a real number. 7. Every point on the line satisfies the equation of the line and every solution of the equation is a point on the line. 8. The solution of a linear equation is not effected when: (i) The same number is added or subtracted from both the side of an equation. (ii) Multiplying or dividing both the sides of the equation by the same non zero number. 2 Top Diagrams 1. Graph of a line passing through the origin. 2. Graph of a line parallel to x axis. 3 3. Graph of a line parallel to y axis. 1 IX Mathematics Ch 6: Lines and Angles Chapter Notes Top Definitions 1. A line segment is a part of a line which has two end points. 2. A ray is a part of a line which has only one end point. 3. A line is a breadth less length which has no end point. 4. Three or more points when lie on the same line are called collinear points. 5. Three or more points when don’t lie on a straight line are called non collinear points. 6. An angle is formed when two rays originate from the same end point. 7. The rays making an angle are called the arms of the angle. 8. The end point from the two rays forming the angle originate is called the vertex of the angle. 9. Two angles whose sum is 90° are called complementary angles. 10. Two angles whose sum is 180° are called supplementary angles. 11. Two angles are adjacent, if they have a common vertex, a common arm and their non–common arms are on different sides of the common arm. 12. If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and vice – verse. This property is called as the linear pair axiom. 13. The vertically opposite angles formed when two lines intersect each other. There are two pairs of vertically opposite angles. 2 14. A line which intersects two or more lines at distinct points is called a transversal. a Corresponding angles: (i) 1 and 5 (ii) 2 and 6 (iii) 4 and 8 (iv) 3 and 7 (ii) 3 and 5 (ii) 2 and 8 b (i) c (i) Alternate interior angles: 4 and 6 Alternate exterior angles: 1 and 7 d Interior angles on the same side of the transversal: (i) 4 and 5 (ii) 3 and6 Top Concepts 1. If a ray stands on a line, then the sum of two adjacent angles so formed is 180°. 3 2. If the sum of two adjacent angles is 180°, then the non – common arms of the angles form a line. 3. If two lines intersect each other, then the vertically opposite angles are equal. 4. If a transversal intersects two parallel lines, then a. Each pair of corresponding angles is equal. b. Each pair of alternate interior angles is equal. c. Each pair of interior angles on the same side of the transversal is supplementary. 5. If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel. 6. If two lines are parallel to the same line, will they be parallel to each other. 7. Lines which are parallel to the same line are parallel to each other. 8. The sum of the angles of a triangle is 180°. 9. If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. 10. In exterior angle of a triangle is greater then either of its interior opposite angles. 11. If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles. Top Diagrams 1. A line 4 2. A ray 3. A line segment 4. Intersecting and non intersecting lines. (i) Intersecting lines 5. (ii) Non–intersecting (parallel) lines ABD and DBC are linear pair of angles 5 6. Types of Angles 1 IX Mathematics Chapter 7: Triangles Chapter Notes Top Definitions 1. Two figures are congruent, if they are of the same shape and of the same size. 2. Two figures are similar, if they are of the same shape but of different size. 3. SAS congruence rule: Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle. 4. ASA congruence rule: Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle. 5. AAS congruence rule: Two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. 6. SSS congruent rule: If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent. 7. RHS congruence rule: If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent. 8. A triangle in which two sides are equal is called an isosceles triangle. Top Concepts 1. If two triangles ABC and PQR are congruent under the corresponding A ↔ P, B ↔Q and C ↔ R, then symbolically, it is expressed as ABC PQR. 2. Two circles of the same radii are congruent. 3. Two squares of the same sides are congruent. 4. Each angle of an equilateral triangle is of 60°. 5. In congruent triangles corresponding parts are equal and we write this as ‘CPCT’ for corresponding parts of congruent triangles. 6. SAS congruence rule holds but not ASS or SSA rule. 7. Angles opposite to equal sides of an isosceles triangle are equal. 2 8. The sides opposite to equal angles of a triangle are equal. 9. RHS stands for Right Angle – Hypotenuse – Side. 10. If two sides of a triangle are unequal, then the greater angle is opposite to the greater side. 11. If two angles of a triangle are unequal, the greater side is opposite to the greater angle. 12. The sum of any two sides of a triangle is greater than the third side. 13. The difference between any two sides of a triangle is less than the third side. 14. If the sum of two adjacent angles is 180°, then the non – common arms of the angles form a line. Top Diagrams 1. ABC DEF 2. ABD DEF 1 IX Mathematics Chapter 8: Quadrilaterals Chapter Notes Top Definitions 1. A quadrilateral is a closed figure obtained by joining four points (with no three points collinear) in an order. 2. A diagonal is a line segment obtained on joining the opposite vertices. 3. Two sides of a quadrilateral having no common end point are called its opposite sides. 4. Two angles of a quadrilateral having common arm are called its adjacent angles. 5. Two angles of a quadrilateral not having a common arm are called its opposite angles. 6. A trapezium is quadrilateral in which one pair of opposite sides are parallel. 7. In the non – parallel sides of trapezium are equal, it is known as isosceles trapezium. 8. A parallelogram is a quadrilateral in which both the pairs of opposite sides are parallel. 9. A rectangle is a quadrilateral whose each angle is 90° 10. A rhombus is quadrilateral whose all the sides are equal. 11. A square is a quadrilateral whose all sides are equal and each angle is 90°. 12. A kite is a quadrilateral in which two pairs of adjacent sides are equal. Top Concepts 1. Properties of parallelogram: i The opposite sides of a parallelogram are parallel. ii A diagonal of a parallelogram divides it in two congruent triangles. iii The opposite sides of a parallelogram are equal. iv The opposite angles of a parallelogram are equal. v The consecutive angles (conjoined angles) of a parallelogram are supplementary. 2 vi The diagonals of a parallelogram bisect each other. 2. A diagonal of a parallelogram divides the parallelogram into two congruent triangles. 3. In a parallelogram opposite sides are equal. 4. If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram. 5. In a parallelogram opposite angles are equal. 6. If in quadrilateral, each pair of opposite angles is equal, then it is a parallelogram. 7. The diagonals of a parallelogram bisect each other. 8. If the diagonals of a quadrilateral bisect other, then it is a parallelogram. 9. A quadrilateral is a parallelogram, if a pair of opposite sides is equal and parallel. 10. Square, rectangle and rhombus are all parallelograms. 11. Kite and trapezium are not parallelogram. 12. A square is a rectangle. 13. A square is a rhombus. 14. A parallelogram is a trapezium. 15. Every rectangle is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rectangle are: 16. i All the (interior) angles of are rectangle are right angles. ii The diagonals of a rectangle are equal. Every rhombus is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rhombus are: i All the sides of rhombus are equal. ii The diagonals of a rhombus intersect at right angles. iii The diagonals bisect the angles of a rhombus. 17. Every square is a parallelogram; therefore, it has all the properties of a parallelogram. Additional properties of a rhombus are: i All the sides are equal ii All the angles are equal to 90° each iii Diagonals are equal 3 iv Diagonal bisect each other at right angle v Diagonals bisects the angles of vertex 18. Sum of all the angles of a quadrilateral is 3600. 19. Mid Point Theorem (Basic Proportionality Theorem): The line segment joining the mid point of any two sides of a triangle is parallel to the third sides and equal to half of it. 20. Converse of mid-point theorem: The line drawn through the mid-point of one side of a triangle parallel to the another side, bisects the third side. 21. If there are three or more parallel lines and the interests made by them on a transversal are equal, then the corresponding intercepts on any other transversal are also equal. 22. A quadrilateral formed by joining the mid-points of the sides of a quadrilateral, in order is a parallelogram. Top Diagrams 1. A quadrilateral ABCD. 2. A trapezium ABCD with sides AB || DC and non parallel sides AD and BC. 3. A parallelogram ABCD in which AB||DC and AD||BC. 4 a 4. A rectangle ABCD with AD||BC, AB||DC and A = 90° = B = C = D. 5. A rhombus ABCD with AB = BD = CD = DA. 6. A square ABCD in which AB = BC = CD, = DA and A = B = C = D = 90°. 7. A kite ABCD with AB = AD and BC = CD 5 8. 9. The relations between special parallelograms can be represented by a Veendiagram. 1 IX Mathematics Chapter 9: Area of Parallelograms and Triangles Quadrilaterals Chapter Notes Top Definitions 1. Any side of a parallelogram is called the base. 2. The length of perpendicular drawn from any point form the parallel sides to the base is called the (corresponding) altitude or height. 3. The part of the plane enclosed by a simple closed figure is called a planar region corresponding to that figure. 4. The magnitude or measure of that planar region is called its area. 5. Two figures are called congruent, if they have the same shape and the same size. 6. Area of a figure is a number (in same unit) associated with the part of the plane enclosed by the two properties. Top Concepts 1. If two figures A and B are congruent, they must have equal areas. 2. Two figures having equal areas need not be congruent. 3. If a planner region formed by a figure T is mad up of two non – overlapping planner regions formed by figures P and Q, then ar(T) = ar(P) + ar(Q). 4. Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices (or the vertex) opposite to the common base of each figure lie on a line parallel to the base. 5. Parallelograms on the same base and between the same parallels are equal in area. 2 6. Area of a parallelogram is the product of its any side and the corresponding altitude. 7. Parallelograms on the same base or equal bases and between the same parallels are equal in area. 8. Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels. 9. Two triangles on the same base (or equal base) and between the same parallel are equal in area. 10. Area of triangle is half the product of its base (or any side) and the corresponding altitude (or height). 11. Two triangles with same base (or equal bases) and equal areas will have equal corresponding altitudes. 12. Two triangles having the same base (or equal bases) and equal areas lie between the same parallels. 13. Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels. 14. A median of a triangle divides it into triangles of equal areas. Top Diagrams 1. Congruent Figures 2. Parallelograms on the same base and between the same Parallels 3 3. Triangles on the same base and between the same parallels 1 IX Mathematics Chapter 10: Circles Chapter Notes Top Definitions 1. A circle is a collection (set) of all those points in a plane, each one of which is at a constant distance from a fixed point in the plane. 2. The fixed point is called the centre and the constant distance is called the radius of the circle. 3. All the points lying inside a circle are called its interior points and all those points which lie outside the circle are called its exterior points. 4. The collection (set) of all interior points of a circle is called the interior of the circle while the collection of all exterior points of a circle is called the exterior of the circle. 5. A line segment joining two points on a circle is called the chord of the circle. 6. A chord passing through the center of the circle is called a diameter of the circle. 7. A line which meets a circle in two points is called a secant of the circle. 8. A polygon is a closed figure made up of three or more line segments (sides) such that each line segment intersects exactly two others at its end – points (vertices) and no two line segments which intersect are collinear. 9. A polygon is called a regular polygon, if it has all its sides equal and has all its angles equal. 10. A (continuous) part of a circle is called an arc of the circle. The arc of a circle is denoted by the symbol ‘ ’. 11. Circumference: The whole arc of a circle is called the circumference of the circle. 12. Semi- circle: One – half of the whole arc of a circle is called a semi – circle of the circle. 13. Minor and Major arcs: An arc less than one - half of the whole arc of a circle is called a minor arc of the circle, and an arc greater than one – half of the whole arc of a circle is called a major arc of the circle. 14. Central Angle: Any angle whose vertex is centre of the circle is called a central angle. 2 15. Degree measure of an Arc: The degree measure of a minor arc is the measure of the central angle subtended by the arc. 16. Congruent Circle: Two circles are said to be congruent if and only if either of them can be superposed on the other so as to cover it exactly. 17. Congruent Arc: Two arcs of a circle (or of congruent) circles) are congruent if either of them can be superposed on the other so as to cover it exactly. 18. Sector of a circle: The part of the plane region enclosed by an arc of a circle and its two bounding radii is called a sector of a circle. 19. Segment of a circle: A chord of a circle divides it into two parts. Each part is called a segment. 20. The part containing the minor arc is called the minor segment, and the part containing the major arc is called the major segment. 21. A quadrilateral, all the four vertices of which lie on a circle is called a cyclic quadrilateral. The four vertices A, B, C and D are said to be Concyclic points. Top Concepts 1. A diameter of circle is its longest chord. 2. A line can meet a circle at the most in two points. 3. In a circle, perpendicular from the center to a chord bisects the chord. 4. In a circle, the line joining the mid – point of a chord to the centre is perpendicular to the chord. 5. Equal chords of a circle are equivalent from the centre of the circle. 6. In a circle, chords which subtend equal angles at the centre are equal. 7. The two points of intersections determine a chord of the circle. 8. In a circle, equal chords subtend equal angles at the centre. 9. In a circle, chords which subtend equal angles at the centre are equal. 10. Triangle is a polygon with 3 sides. 11. Quadrilateral is a polygon with 4 sides. 12. The chords corresponding to congruent arcs are equal. 3 13. If two arcs of a circle (or of congruent circles) are congruent, then the corresponding chords are equal. 14. If two chords of a circle (or of congruent circles) are equal, then their corresponding arcs (minor, major or semi – circular) are congruent. 15. One and only one circle can be drawn through three non – collinear points. 16. An infinite number of circles can be drawn through a given point P. 17. An infinite number of circles can be drawn through the two given points. 18. Perpendicular bisectors of two chords of a circle, intersect each other at the centre of the circle. 19. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. 20. Angles in the same segment of a circle are equal. 21. An angle in a semi–circle is a right angle. 22. The arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semi–circle. 23. If a line segment joining two points subtends equal angles at two other points lying on the same side of the line segment, the four points are concyclic, i.e., lie on the same circle. 24. An angle in a semi–circle is a right angle. 25. The arc of a circle subtending a right angle at any point of the circle in its alternate segment is a semi–circle. 26. If a line segment joining two points subtends equal angles at two other points lying on its same side of the line segment, the four points are concyclic i.e., lie on the same circle. 27. If the sum of any pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic. 28. Any exterior angle of a cyclic quadrilateral is equal to the interior opposite angle. Top Formulae 1. Diameter = 2 x Radius. 2. If the degree measure of AB is θ°, we write m AB is θ°. 4 3. The degree measure of a semi – circles is 180° 4. The degree measure of a circle is 360°. 5. The degree measure of a major arc is (360° - θ°), where θ° is the degree measure of the corresponding minor arc. 6. For a quad. ABCD, A + C = 180° or B = D = 180°, then ABCD is cyclic. 7. Area of a circle = r2 Top Diagrams 1. Interior and Exterior of a Circle 2. Concentric circles 3. Secant, Diameter and Chord in a circle. 5 4. 5. 6. 7. Arc of a circle Circumference of a circle Semi-Circle Minor and Major arc 6 8. Minor and Major Sector 8. Minor and Major Segment 9. 10. Circles passing through a point. Circles passing through two points. 7 11. 12. Chord bisectors meet at center. Cyclic Quadrilateral 1 IX Mathematics Chapter 11: Geometric Constructions Chapter Notes Top Concepts 1. To construct an angle equal to a given angle. Given : Any POQ and a point A. Required : To construct an angle at A equal to POQ. Steps of Construction: 1. With O as centre and any (suitable) radius, draw an arc to meet OP at R and OQ at S. 2. Through A draw a line AB. 3. Taking A as centre and same radius (as in step 1), draw an arc to meet AB at D. 4. Measure the segment RS with compasses. 5. With d as centre and radius equal to RS, draw an arc to meet the previous arc at E. 6. Join AE and produce it to C, then BAC is the required angle equal to POQ 2. To bisect a given angle. Given : Any POQ Required : To bisect POQ. Steps of Construction: 1. With O as centre and any (suitable) radius, draw an arc to meet OP at R and OQ at S. 2 2. With R as centre and any suitable radius (not necessarily) equal to radius of step 1 (but > 1 2 RS), draw an arc. Also, with S as centre and same radius draw another arc to meet the previous arc at T. 3. 3. Join OT and produce it, then OT is the required bisector of POQ. To construct angles of 60°, 30°, 120°, 90°, 45° (i) To construct an angle of 60° Steps of Construction: 1. Draw any line OP. 2. With O as centre and any suitable radius, draw an arc to meet OP at R. 3. With R as centre and same radius (as in step 2), draw an arc to meet the previous arc at S. 4. Join OS and produce it to Q, then POQ = 60°. 3 (ii) To construct an angle of 30° Steps of Construction 1. Construct POQ = 60° (as above). 2. Bisect POQ (as in construction 2). Let OT be the bisector of POQ, then POT = 30° (iii) To construct an angle of 120° 1. Draw any line OP. 2. With O as centre and any suitable radius, draw an arc to meet OP at R. 3. With R as centre and same radius (as in step 2), draw an arc to meet the previous arc at T. With T as centre and same radius, draw another arc to cut the first arc at S. 4. Join OS and produce it to Q, then POQ = 120°. 4 (iv) To construct an angle of 90° Steps of Construction 1. Construct POQ = 60° (as in construction 3(i)). 2. Construct POV = 120° (as above). 3. Bisect QOV (as in construction 2). Let OU be the bisector of QOV, then POU = 90°. (v) To construct an angle of 45° Steps of Construction 1. Construct AOP = 90° (as above). 2. Bisect AOP (as in construction 2). Let OQ be the bisector of AOP, then AOQ = 45° 5 4. To bisect a given line segment. Given : Any line segment AB. Required : To bisect line segment AB. Steps of Construction: 1. At A, construct any suitable angle BAC. 2. At B, construct ABD = BAC on the other side of the line AB. 3. With A as centre and any suitable radius, draw an arc to meet AC at E. 4. From BD, cut off BF = AE. 5. Join EF to meet AB at G, then EG is a bisector of the line segment AB and G is mid – point of AB. (ii) To divided a given line segment in a number of equal part. 6 5. Divided a line segment AB of length 8 cm into 4 equal part. Given : A line segment AB of length 8 cm. Required : To divide line segment 8 cm into 4 equal parts. Steps of Construction: 1. Draw lien segment AB = 8 cm. 2. At A, construct any suitable angle BAX. 3. At B, construct ABY = BAX on the other side of the line AB. 4. From AX, cut off 4 equal distances at the points C, D, E and F such that AC = CD = DE = EF. 5. With the same radius, cut off 4 equal distances along BY at the points H, I, J and K such that BH = HI = IJ = JK. 6. Join AK, CJ, DI, EH and FB. Let CJ, DI and EH meet the line segment AB at the points M, N and O respectively. Then, M, N and O are the points of division of AB such that AM = MN = NO = OB. 6. To draw a perpendicular bisector of a line segment. Given : Any line segment PQ. Required : To draw a perpendicular bisector of lien segment PQ. Steps of Construction: 7 1. With P as centre and any line suitable radius draw arcs, one on each side of PQ. 2. With Q as centre and same radius (as in step 1), draw two more arcs, one on each side of PQ cutting the previous arcs at A and B. 3. Join AB to meet PQ at M, then AB bisects PQ at M, and is perpendicular to PQ, Thus, AB is the required perpendicular bisector of PQ. 7. To construct an equilateral triangle when one of its side is given. E.g.: Construct and equilateral triangle whose each side is 5 cm. Given : Each side of an equilateral triangle is 5 cm. Required : To construct the equilateral triangle. Steps of Construction: 1. Draw any line segment AB = 5 cm. 2. With A as centre and radius 5 cm draw an arc. 3. With B as centre and radius 5 cm draw an arc to cut the previous arc at C. 4. Join AC and BC. Then ABC is the required triangle. 8 8. To construct an equilateral triangle when its altitude is given. E.g.: Construct an equilateral triangle whose altitude is 4 cm. Steps of Construction: 1. Draw any line segment PQ. 2. Take an point D on PQ and At D, construct perpendicular DR to PQ. From DR, cut off DA = 4 cm. 3. At A, construct DAS = DAT = 1 2 60 = 30° on either side of AD. Let AS and AT meet PQ at points B and C respectively. Then, ABC is the required equilateral triangle. 9. Construction of a triangle, given its Base, Sum of the other Two sides and one Base Angle. 9 E.g Construct a triangle with base of length 5 cm, the sum of the other two sides 7 cm and one base angle of 60°. Given: In ΔABC, base BC = 5 cm, AB + AC = 7 cm and ABC = 60° Required : To construct the ΔABC. Steps of Construction: 1. Draw BC = 5 cm. 2. At B, construct CBX = 60° 3. From BX, cut off BD = 7 cm. 4. Join CD. 5. Draw the perpendicular bisector of CD, intersecting BD at a point A. 6. 10. Join AC. Then, ABC is the required triangle. Construction of a triangle, Given its Base, Difference of the Other Two Sides and one Base Angle. Eg: Construct a triangle with base of length 7.5 cm, the difference of the other two sides 2.5 cm, and one base angle of 45° Given : In ΔABC, base BC = 7.5 cm, the difference of the other two sides, AB – AC or AC – AB = 2.5 cm and one base angle is 45°. Required : To construct the ΔABC, CASE (i) AB – AC = 2.5 cm. Steps of Construction: 10 1. Draw BC = 7.5 cm. 2. At B, construct CBX = 45°. 3. From BX, cut off BD = 2.5 cm. 4. Join CD. 5. Draw the perpendicular bisector RS of CD intersecting BX at a point A. 6. Join AC. Then, ABC is the required triangle. CASE (ii) AC – AB = 2.5 cm Steps of Construction: 1. Draw BC = 7.5 cm. 2. At B, construct CBX = 45° and produce XB to form a line XBX’. 3. From BX’, cut off BD’ = 2.5 cm. 4. Join CD’. 5. Draw perpendicular bisector RS of CD’ intersecting BX at a point A. 6. Join AC. Then, ABC is the required triangle. 11 e 11. Construction of a Triangle of Given Perimeter and Base Angles. Construct a triangle with perimeter 11.8 cm and base angles 60° and 45°. Given : In ΔABC, AB+BC+CA = 11.8 cm, B = 60° & C = 45°. Required : To construct the ΔABC. Steps of Construction: 1. Draw DE = 11.8 cm. 2. At D, construct EDP = DEQ = 1 2 1 of 45° = 22 2 of 60° = 30° and at E, construct 1 2. 3. Let DP and EQ meet at A. 4. Draw perpendicular bisector of AD to meet DE at B. 5. Draw perpendicular bisector of AE to meet DE at C. 6. Join AB and AC. Then, ABC is the required triangle. 12 1 IX Mathematics Chapter 12: Heron’s Formula Chapter Notes Top Definitions 1. The region enclosed with in a simple closed figure is called its area. 2. A plane figure bounded by four sides is a quadrilateral. 3. A quadrilateral is a cyclic quadrilateral if all its four vertices lie on the circumference of the circle. 4. Semi perimeter is half of the perimeter. Top Concepts 1. For every triangle, the values of (s – a), (s – b), and (s – b) are positive. 2. The line segment joining the mid-point to any of the vertex divides the triangle in two parts, equal in area. 3. The diagonal of a quadrilateral divides the quadrilateral into two triangles. 4. The diagonal of a parallelogram divides the quadrilateral into two congruent triangles. 5. Area of a quadrilateral whose sides and one diagonal are given can be calculated by dividing the quadrilateral into two triangles and using Heron’s formula. Top Formulae 1. In triangle ABC right angled at B, AB2 + BC2 = AC2 2. Area of equilateral triangle = 4 length of an equilateral triangle. 3. Semi-perimeter of equilateral triangle = 4. Area of a triangle = 3 1 a2 sq units, where ‘a’ is the side 2 base height 3a 2 2 5. Area of triangle = s(s - a)(s - b)(s - c) , s semi perimeter 6. Area of parallelogram = base × height 7. Area of a triangle = 8. Area of parallelogram = 2 x (Area of triangle) 9. Area of cyclic quadrilateral = s(s - a)(s - b)(s - c)(s-d) a + b + c+d s semi perimeter 2 10. Area of a rhombus = 11. Area of a trapezium = 12. Area of a quadrilateral = 1 2 1 2 base height 1 2 Pr oduct of diagonals 1 2 height x(sumof parallelsides) diagonal x sumof perpendicular fromvertices ondiagonal a+b+c 2 1 Class IX: Math Chapter 14: Statistics Chapter Notes Top Definitions 1. Facts or figures collected with a definite purpose are called data. 2. Statistics deals with collection, presentation, analysis and interpretation of numerical data. 3. Arranging data in a order to study their salient features is called presentation of data. 4. Data arranged in ascending or descending order is called arrayed data or an array. 5. When an investigator with a definite plan or design in mind collects data first handedly, it is called primary data. 6. Data when collected by someone else, say an agency or an investigator, comes to you, is known as the secondary data. 7. Variable is a quantity that assumes different values. 8. Range of the data is the difference between the maximum and the minimum values of the observations. 9. The small groups obtained on dividing all the observations are called classes or class intervals and the size is called the class size or class width. 10. Class mark of a class is the mid value of the two limits of that class. 11. A bar graph is the diagram showing a system of connections or interrelations between two or more things by using bars. 12. A histogram is the bar graph such that the area over each class interval is proportional to the relative frequency of data within this interval. 13. The number of times an observation occurs in the data is called the frequency of the observation. 14. A frequency distribution in which the upper limit of one class differs from the lower limit of the succeeding class is called an Inclusive or discontinuous Frequency Distribution. 15. A frequency distribution in which the upper limit of one class coincides from the lower limit of the succeeding class is called an exclusive or continuous Frequency Distribution. 2 16. A bar graph is a pictorial representation of data in which rectangular bars of uniform width are drawn with equal spacing between them on one axis, usually the x axis. The value of the variable is shown on the other axis that is the y axis. 17. A histogram is a set of adjacent rectangles whose areas are proportional to the frequencies of a given continuous frequency distribution. 18. The Cumulative Frequency of a class-interval is the sum of frequencies of that class and the classes which precede (come before) it. 19. The mean value of a variable is defined as the sum of all the values of the variable divided by the number of values. 20. Median is the value of middle most observation(s). 21. Mode of a statistical data is the value of that variate which has the maximum frequency. Top Concepts 1. In case of continuous frequency distribution, the upper limit of a class is not to be included in that class while in discontinuous both the limits are included. 2. The height of rectangles corresponds to the numerical value of the data. 3. Frequency polygons are a graphical device for understanding the shapes of distributions. 4. Bar charts are used for comparing two or more values. 5. A histogram differs from a bar chart, as in the former it is the area of the bar that denotes the value, not the height. 6. The height of the rectangle as the ratio of the frequency of the class to the width or size of the class. 7. Last cumulative frequency is always the sum total of all the frequencies. 8. If both a histogram and a frequency polygon are to be drawn on the same graph, then we should first draw the histogram and then join the mid-points of the tops of the adjacent rectangles in the histogram with line-segments to get the frequency polygon. 9. If classes are not of equal width, then the height of the rectangle is calculated by the ratio of the frequency of that class, to the width of that class. 3 10. A measure of central tendency tries to estimate the central value which represents the entire data. 11. The three measures of central tendency for ungrouped data are mean, mode and median. 12. The disadvantage of arithmetic mean is that it is affected by extreme values. 13. The median is to be calculated only after arranging the data in ascending order or descending order. 22. Average height is the modal value. 23. Disadvantage of the mode is that it is not uniquely defined in many cases. 24. The data is symmetric about the mean position when the three averages mean median and mode are all equal. 25. The data is asymmetric when the three measures are unequal. 14. The variate corresponding to the highest frequency is to be taken as the mode and not the frequency. Top Formulae Range 1. Class size = Number of classes 2. Class size = Upper limit – Lower Limit 3. Mean ( x ) 1 n n xi i1 4. Mean ( x) f i xi f i 5. (i) If number of observations (n) is odd, Median = ( n 1 )th observation 2 n th n ( 2 ) ( 2 1)th (ii) If n is even, then median observation 2 4 Top Diagrams 1. Symmetric Distribution 2. Asymmetrical or skewed distribution 5 3. Bar Graph 4. Mean < Mode 6 5. Mode < Mean 6. Frequency Polygons 7 7. A histogram