CHAPTER 14 VISUALISING SOLID SHAPES
2-D figures are the plane figures and 3-D figures are the solid shapes.
The corners of a solid shape are called its vertices. The line segments of are called its edges. The flat surfaces are called its faces.
Two types of sketches of solids are possible:
1. An oblique sketch does not have proportional lengths.
2. An isometric sketch is drawn on an isometric dot paper. The measurements are kept proportional.
Different sections of a solid can be viewed in many ways:
1. Cutting or slicing the shape i.e. cross section
2. 2-D shadow of a 3-D shape
3. Looking at the shape from different angles: the front-view, the side-view, the
CHAPTER 13 SYMMETRY
A figure has a line symmetry if there is a line about which the figure may be folded so that the two parts of the figure will coincide.
Number of lines of symmetry of:
1. Regular hexagon= 6
2. Regular pentagon= 5
3. Square= 4
4. Equilateral triangle= 3
The fixed point about which a figure is rotated is called its centre of rotation.
The angle by which the object is rotated is called the angle of rotation.
After rotation, if the object looks exactly the same, it is said to have rotational symmetry.
In a complete 3600 turn, the number of times an object looks exactly the same is called the order of rotational symmetry.
CHAPTER 12 EXPONENTS AND POWERS
Very large numbers are difficult to read, understand, compare and operate upon. To make all these easier, we use exponents, converting many of the large numbers in a shorter form.
Numbers in exponential form obey certain laws, which are:
For any non-zero integers a and b and whole numbers m and n,
CHAPTER 11 ALGEBRAIC EXPRESSIONS
Terms are added to form an expression. A term is a part of an expression which is formed first and then added.
A term is a product of the factors.
The numerical factor in a term is called the coefficient.
LIKE AND UNLIKE TERMS
Terms having same algebraic factors are called like terms.
Terms having different algebraic factors are called unlike terms.
TYPES OF EXPRESSIONS
Monomial: One term expression
Binomial: Two term expression
Trinomial: Three term expression
Polynomial: One or more term expression
SUM OR DIFFERENCE
The sum or difference of two like terms is a like term with coefficient equal to the sum or difference of the coefficients of the two like terms.
When we add two unlike terms, they are left as they are.
1. Square= 4 x side
2. Rectangle= 2 x (length + breadth)
3. Parallelogram and triangle= sum of the sides
4. Circle= 2 x π x radius
1. Square= side x side
2. Rectangle= length x breadth
3. Parallelogram= base x height
4. Triangle= x base x height
5. Circle= π x (radius)2
CHAPTER 9 RATIONAL NUMBERS
A rational number is a number that can be expressed in the form of where p and q are integers and q≠0.
EQUIVALENT RATIONAL NUMBERS
By multiplying the numerator and denominator of a rational number by the same non-zero integer, we obtain another rational number equivalent to the given rational number.
RATIONAL NUMBERS IN STANDARD FORM
A number is said to be in standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1.
COMPARISON OF RATIONAL NUMBERS
Make the denominators equal and compare the numerators. The one with the greater numerator is the greater rational number.
OPERATIONS OF RATIONAL NUMBERS
Operations on rational numbers are done just like fractions.
CHAPTER 8 COMPARING QUANTITIES
To compare two quantities, their units must be the same.
If converting the ratios into fractions and then making them like fractions turns them into equal fractions, the ratios are equivalent ratios.
Percentage is how much the quantity is a part of a hundred.
CONVERTING FRACTIONS INTO PERCENTAGE
To convert fractions into percentage, multiply the fraction by 100.
CONVERTING DECIMALS INTO PERCENTAGE
Convert the decimal into fraction and then multiply it by 100.
CHANGE IN PERCENTAGE
To find out the change in any quantity in percentage,
% change = (Amount of change)/(Original amount) x 100
BUYING AND SELLING
The buying price of any item is called its cost price (CP). The price at which it is sold is called its selling price (SP).
Profit is when the SP is more than the CP.
Loss is when the CP is more than the SP.
If CP=SP, there is no profit, no loss.
% profit = Profit/CP x 100
% loss = Loss/( CP) x 100
The borrowed amount is called the principal.
The extra amount paid along with the principal is called the interest.
Simple Interest = (Principal x Rate x Time)/100
CHAPTER 7 CONGRUENCE OF TRIANGLES
If two figures are exactly alike in shape and size, they are said to be congruent.
This property of being congruent is called congruence.
CONGRUENCE OF LINE SEGMENTS
Two line segments having same length are said to be congruent.
CONGRUENCE OF ANGLES
Two angles having same the measure are said to be congruent angles.
CONGRUENCE OF TRIANGLES
Two triangles are said to be congruent if their corresponding angles and sides are equal.
Criteria of congruence of triangles:
1. SSS: Three sides of a triangle are equal to the corresponding three sides of the other triangle.
2. SAS: Two sides and the angle between them are equal to the corresponding two sides and the angle between them.
3. ASA: Two angles and the side between them are equal to the corresponding angles and the side between them.
Congruence of right-angled triangle:
RHS: If the hypotenuse and one side of a right-angled triangle are equal to the right angle and hypotenuse of the other right-angled triangle, they are said to be congruent.
A median connects the vertex of a triangle to the mid-point of the opposite side.
An altitude is a perpendicular line segment from one vertex to the opposite side of the triangle.
An exterior angle of a triangle is equal to the sum of its interior opposite angles.
A triangle having all three sides equal is called an equilateral triangle.
All three angles of an equilateral triangle are equal to 60o.
A triangle having two equal sides is called an isosceles triangle.
In an isosceles triangle, base angles opposite to the equal sides are equal.
SUM OF TWO SIDES
The sum of the length of any two sides is greater the length of the third side.
The difference between the lengths of any two sides is smaller than the length of the third side.
RIGHT ANGLED TRIANGLE
In a right angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called legs.
According to Pythagoras theorem, the square of the hypotenuse is equal the sum of the squares of the other two sides.
CHAPTER 5 LINES AND ANGLES
An angle is formed when two lines meet.
Complementary angles: Two angles which add up to 900 are called complementary angles.
Supplementary angles: Two angles which add up to 180o are called supplementary angles.
Adjacent angles: Angles which have a common vertex and a common arm but no common interior.
Linear pair: Angles which are adjacent and supplementary
When two lines meet, they are said to intersect with each other. The point at which they meet is called the intersection point.
When two or more lines do not intersect, no matter how far they are extended, they are said to be parallel lines.
VERTICALLY OPPOSITE ANGLES
When two lines intersect like an X, the pair of opposite angles are called vertically opposite angles. Vertically opposite angles are equal.
A transversal line is a line that intersects two or more lines.
When a transversal cuts two parallel lines, there are some properties like: