Important points and formulae
1. Percentages:
Increase = new value – original value
Decrease = original value – new value
Profit = selling price – cost price
Loss = cost price – selling price
Discount = marked price – sale price
discount
 100 %
markedpric e
profit / loss
 100 %
% profit/ loss =
originalpr ice
% discount =
2. Congruence and similarity
Two triangles are congruent if
a) All sides of one triangle are equal to all sides of another triangle. (SSS property)
b) Two sides and including angle of one triangle are equal to two corresponding
sides and including angle of another triangle. (SAS property)
c) Two angles and one side of one triangle are equal to two corresponding angles
and one side of another triangle. (AAS property)
d) A figure and its image under translation, rotation and reflection are congruent.
When two polygons are similar, then
a) All corresponding angles are equal,
b) All corresponding sides are proportional.
c) A figure and its image under an enlargement are similar.
d) An enlargement with a scale factor greater than 1 produces an enlarged figure.
An enlargement with a scale factor between 0 and 1 produces a diminished image.
An enlargement with a scale factor 1 produces a congruent image.
e) Two triangles are said to be similar if,
All three angles of one triangle are equal to all corresponding angles of another
triangle. (AA property)
One angle of both the triangles is equal and two corresponding sides are
proportional. (ASS property)
f) If two shapes are similar,
A1
l1
 ( )2
A2
l2
g) If two shapes are similar,
v1
l1
 ( )3
v2
l2
h) If XY is parallel to BC then
AX/ XB = AY / YC (see fig)
4. Algebraic identities
a) (a+b)2 = a2 + 2ab + b2 ,
b) (a-b)2 = a2 – 2ab + b2 ,
c) (a+b)(a-b) = a2 – b2
Quadratic formula
x
b
b 2  4ac
2a
5. Pythagoras theorem
For any right angled triangle
Hypotenuse (h2) = base (b2) + perpendicular (p2)
 The converse of the Pythagoras theorem states that if h2 = b2 + p2 then the triangle
is a right angled triangle.
 Hypotenuse is always greater than a base or a perpendicular.
 Height of any polygon is always a perpendicular distance.
6. Mensuration:
Area, perimeter of two dimensional shapes.
Shape
perimeter
area
square
4 length
Length2
rectangle
2(L+B)
L×B
Triangle
Sum of all three sides
½× base×height
parallelogram
Trapezium
2(L+B)
Sum of all four sides
base×height
½× base(sum of parallel sides)
Volume and surface area of three dimensional shapes
shape
Cube
Cuboids
Cylinder
Prism
Pyramid
Volume
Length3
L×B×H
πr2h
Base area
×height
1/3 Base area
×height
1/3 πr2h
4/3 πr3
πR2h – πr2h
Surface area
6 length2
2(L×B) +2(L×H)+2(B×H)
2πr2+ 2πrh
(Perimeter of the base) ×h + 2(base area)
Area of the base + area of the triangles on the
surface.
πr2+ πrl (curved surface area)
4 πr2
External curved surface area+ internal curved
surface area + area of 2 rings
2 πr2 + πr2
Cone
sphere
Hollow
cylinder
hemisphere
2/3 πr3
7. Circle:
Area of circle = πr2
Perimeter of circle = circumference = 2 πr
Area of ring = πR2 – πr2
Length of arc = x0/ 360 × 2 πr
Area of sector = x0/ 360 × πr2
Perimeter of sector = length of arc + 2 radius
Area of segment = Area of sector – area of triangle
1 revolution (in terms of angle) = 3600
1 revaluation(in terms of distance) = circumference= 2 πr
8. Coordinate geometry
The distance between the two points (x1,y1) and (x2,y2) is
Distance between two points on a horizontal line.
D = x 2  x1
Distance between two points on a vertical line.
D = y 2  y1
If A,B,and C are collinear points then Gradient of AB = gradient of BC =
gradient of AC.
The equation of a line which has gradient m and which passes through the point
(x1, y1) is:
y - y1 = m(x - x1)
The equation of a line passing through (x1, y1) and (x2, y2) can be written as:
y - y1 = y2 - y1
x - x1 x2 - x1
In the coordinate plane, two lines are perpendicular if the product of their
gradients is -1.
For example: The line
is perpendicular to the line y = –2x – 1. The product of
the two slopes is
In the coordinate plane, two lines are parallel if their gradients are equal.
Heron's Formula for the area of a triangle Also called Hero's Formula
A method for calculating the area of a triangle when you know the lengths of all three
sides.
Let a,b,c be the lengths of the sides of a triangle. The area is given by:
where p is half the perimeter, or
Graph of equation of the form y = c is parallel to x-axis and the gradient is 0.
X = c parallel to y-axis and gradient is undefined.
Y = mx passes through the origin and has the gradient, m
Y = mx + c has gradient m and intersect y-axis at c.
Area =
x1( y 2  y 3)  x 2( y 3  y1)  x3( y1  y 2)
2
To find the gradient of the line, equation of the line must be in the form y= mx+c.
where m is the gradient.
Gradient =
y 2  y1
x 2  x1
Any point on x-axis is (x,0)
Any point on y axis is (0, y)
To find the coordinates of the point of intersection of two straight lines, solve
equations of the two lines simultaneously.
9. Graphs
The general form of Quadratic function is y= ax2 + bx + c
The quadratic graph has the minimum point when ‘a’ is negative. It has a
maximum point when ‘a’ is positive.
The line of symmetry of the quadratic graph passes through the maximum or
minimum point.
16. Laws of Indices:
am × an = am+n
am ÷ an = am-n
(a×b)m = am × bm
(a/b)m = am/ bm
(am)n = amn
a0 = 1
(a/b)-m = (b/a)m
17. Measure of central tendency:
For simple distribution
Mean = sum of numbers
Number of numbers
Mode = The most frequent number
Median = the middle number (if n is odd)
Sum of two middle numbers/ 2 (if n is even)
For frequency distribution
Mean =
 f ( x)
f
Mode = The value of x, which has highest frequency
Median = Add all frequencies, divide by 2, if answer is with decimal round it off
and locate that number in the table. If answer is whole number, take the number
and next number, locate these two numbers add and divide by 2.
19. Matrices:
A+O = O+A = A
Additive inverse of A is denoted by –A
Multiplicative inverse of A is denoted by A-1
A-1A = AA-1 = I where I =
1  0
0  1

AI = IA = A
Sketches of some important graphs