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Geometry and Trigonometry

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INTRODUCTION

1. EUCLID’S GEOMETRY
2. LINES AND ANGLES
3. TRIANGLES
4. QUADRILATERIALS
5. CIRCLES
6. COORDINATE GEOMETRY
7. AREAS OF PARALLELOGRAMS AND TRIANGLES
8. SURFACE AREAS AND VOLUMES

 

1. EUCLID’S GEOMETRY

The geometry of plane figures is known as “Euclid’s Geometry”.

EUCLID’S DEFINITIONS, AXIOMS AND POSTULATES:

A solid has three dimensions, a surface has two, a line has one and a point has none. Euclid summarized these statements as definitions. Some of them are:

(i)   A point is that which has no parts

(ii)  A line is a breadth less length

(iii) The end of a line are points

(iv) A straight line is a line which lies evenly with the points on itself.

(v)  A surface is that which has length and breadth only.

(vi) A plane surface is a surface which lies evenly with straight lines on itself.

Euclid assumed certain property which were not proved. He divided them into two types:

(i)   axioms and   (ii)     postulates

Postulates, were the assumption specific to geometry. Common notions called axioms, were assumptions used throughout Mathematics and not specifically linked to geometry.

BASIC GEOMETRICAL CONCEPTS:

Axioms: The basic facts which are taken for granted, without proof, are called axioms.

(i)   Things which are equal to the same thing are equal to one another.

If a = b, b = c   => a = c

(ii)  The equals are added to equals, the whole are equal

a + c = b + c

(iii) If equals are subtracted from equals, the remainders are equal

If a = b => a – c = b – c

(iv) Things which are double of the same things are equal to one another.

If a = b => 2a = 2b

(v)  Things which are halves of the same things are equal to one another.

If a = b => a/2 = b/2

(vi) Things which are greater than the same thing are greater than one another.

If a > b, b > c => a > c

(vii)       Things which coincide with one another are equal to one another.

(viii)      The whole is greater than the part.

POSTULATES :

Postulate – 1 :

A straight line may be drawn from any one point to any another point.

 

Postulate – 2 :

A terminated line can be produced indefinitely

Postulate – 3 :

A circle can be drawn with any centre and any radius.

where O is the centre of the circle and OA = (r) radius of the circle.

Postulate – 4 :

All right angles are equal to one another

the measurement of Because each and every right angle is always 90o.

Postulate – 5 :

If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

e.g. Line PQ falls on lines AB and CD such that the sum of interior angles ∠1 + ∠2 < 180o is on the left side of PQ. So, the lines AB and CD will eventually intersect on the left side of PQ.

Theorem:  Two distinct lines cannot have more than one point in common given.



EQUIVALENT VERSIONS OF EUCLID’S FIFTH POSTULATE :

Two distinct intersecting lines cannot be parallel to the same line.



Euclid’s fifth postulate is very significant in the history of Mathematics. By implication, we can see that no intersection of lines will take place when the sum of the measures of the interior angles on the same side of the falling line is exactly 180o.

There are several equivalent versions of this postulate. One of them is “Play fair’s Axom” which was given by a scottish Mathematician John play fair in 2729, as stated below:
For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l.

2. LINES AND ANGLES

BASIC TERMS AND DEFINITIONS:

I  A line: When two or more than two points are joined end point, it is called aline. It is denoted by .

II  A line Segment: A part (or portion) of a line with two end points is called a line segment e.g. AB is a line segment and denoted by AB.

III  A ray: A part of line with one end point is called a ray;  e.g. PQ is a ray and denoted by

IV  Collinear points : If three or more points lie on the same line, they are called collinear points. i.e. A, B, C, D, E and F are collinear points.

V  Non-collinear points : If three or more points do not lie on the same line, they are called non-collinear points. i.e. P, Q, R, S, T, U and V are non-collinear points.


VI  An Angle: When two rays originate from the same end point, an angle is formed ; e.g. ∠AOB is an angle and OA and OB are called the arms of an angle ∠AOB. The measurement of an angle is degree.

VII  Vertex: The end point of the arms of an angle is called the vertex of an angle; e.g. O is the vertex of an angle ∠AOB.

KINDS OF ANGLE:

(i)    An acute angle: The angles between 0o and 90o are called acute angles. i.e. 0o < acute angle < 90o;

e.g.          ∠AOB is an acute angle.

(ii)   A right angle: A right angle is exactly equal to 90o, i.e., right angle = 90o

e.g.  ∠POQ is 90o (a right angle)

(iii)  An obtuse angle: An angle greater than 90o but less than 180o is called an obtuse angle,

i.e. 90o < obtuse angle < 180   e.g.  ∠MOP is an obtuse angle.

(iv)  Straight angle: A straight angle is equal to 180o, i.e. a straight angle is 180o or is 2 × 90o = 2 right angles; e.g.  ∠COD is a straight angle.

(v)   Reflex angle: An angle which is greater than 180o but less than 360o is called a reflex angle,

i.e., 180o < reflex angle < 360o; e.g.  ∠EOF is a reflex angle.

(vi)  Complementary angles: Two angles whose sum is 90o are called complementary angles

i.e.,          ∠x + ∠y = 90o

e.g.          ∠ABD + ∠DBC = 90o               [Complementary angles]

40o + 50o = 90o                                   [Complementary angles]

60o + 30o = 90o                                   [Complementary angles]

70o + 20o =90o                                  [Complementary angles]

80o + 10o = 90o                                   [Complementary angles]

45o + 45o = 90o                                   [Complementary angles]


(vii) Supplementary angles: Two angles whose sum is 180o are called supplementary angles

i.e.,   ∠x + ∠y  = 180o                     [Two right angles = 2× 90o = 180o]

e.g.            ∠ABD + ∠DBC = 180o               [supplementary angles]

90o + 90o = 180o                                                      [supplementary angles]

100o + 80o = 180o                                                   [supplementary angles]

110o + 70o = 180o                           [supplementary angles]

120o + 60o = 180o                                                 [supplementary angles]

130o + 50o = 180o                                                    [supplementary angles]

90o + 90o = 180o                                                      [supplementary angles]



(viii)      Adjacent angles: If two angles have a common vertex and a common arm, they are called adjacent angles, i.e. ∠ABD and ∠DBC have common arm BD and also common vertex B, so, they are

djacent angles. e.g. ∠x and ∠y are adjacent angles.

(ix)  Linear pair of angles: If the non common arms QP and QR in the given figure, from a line, then the angles ∠PQS and ∠SQR are called linear pair of angles.

(x)   Vertically opposite angles: When two lines intersect each other at a point, they make two pairs of vertically opposite angles such type of angles are also equals.

e.g.        ∠AOC = ∠BOD       [Vertically opposite angles]

∠COB = ∠AOD                                [Vertically opposite angles]

INTERSECTING LINES AND NON-INTERSECTING LINES:

(i)    Intersecting Lines: If two lines intersect each other at any point, they are called intersecting lines.

e.g. AB and CD are intersecting lines because they intersect each other at a point O.

(ii)   Non-intersecting (parallel) lines: If two lines never intersect each other and the distance between them is always equal (same), they are called non-intersecting (parallel) lines, i.e., parallel lines do not

intersect even at infinity e.g. CD || MN

Theorem:  If two lines intersect each other, then vertically opposite angles are equal.

Parallel Lines and a TRANSVERSAL:

If l || m and t is a transversal.

Then,               (i)  Corresponding angles:

∠1 = ∠5

∠2 = ∠6

∠4 = ∠8

∠3 = ∠7

(ii)  Alternate interior angles:

∠4 = ∠6

∠3 = ∠5

(iii) Alternate exterior angles:

∠1 = ∠7

∠2 = ∠8

(iv) Interior angles on the same side of the transversal:

∠4 + ∠5 = 180o and ∠3 + ∠ 6 = 180o

CORRESPONDING ANGLES AXIOM:

Axiom 1:  If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

If l || m and t is a transversal, then corresponding angles:

∠1 = ∠5

∠2 = ∠6

∠4 = ∠8

∠3 = ∠7

Axiom 2:   If a transversal intersects two lines such that a pair of corresponding angles is equals, then the two lines are parallel to each other.

If transversal PS intersects two lines AB and CD such that

∠AQP = ∠CRQ                                [Pair or corresponding angles]

or ∠BQP = ∠DRQ                            [Pair of corresponding angles]

then, AB || CD


Theorem:        If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. If transversal PS intersects two parallel lines AB and CD respectively,

Then, ∠AQR = ∠QRD                      [Pair of alternate interior angles]

And ∠BQR = ∠CRQ                         [Pair of alternate interior angles]

Theorem:        If a transversal intersect two lines such that a pair of alternate interior angles is equal, then the two lines are parallel

If PS transversal intersect two lines AB and CD such that

∠BQR = ∠CRQ                                [A pair of alternate interior angles]

Then,  B || CD

Theorem:  If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

             If a transversal t intersects two parallel lines AB and CD at P and Q points respectively,

             Then, ∠APQ + ∠CQP = 180o             [Pair of interior angles]

             And ∠BPQ + ∠DQP = 180               [Pair of interior angles]

Theorem: If a transversal intersect two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

If a transversal t intersects two lines AB and CD such that a pair of interior angles on the same side of the transversal is supplementary, i.e.,

             ∠APQ + ∠BPQ = 180o                       [Supplementary]

             And ∠CQP + ∠DQP = 180o                [Supplementary]

             Then,  AB || CD

LINES PARALLEL TO THE SAME LINE

Theorem:  Lines which are parallel to the same line are parallel to each other.


            
Theorem:  The sum of the angles of a triangle is 180o.

Theorem:  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.

            ∠1 + ∠2 = ∠4
             ∠BAC + ∠ABC = ∠ACD

3. TRIANGLES

CONGRUENCE OF TRIANGLES:

Congruent means equal in all the respect or geometrical figures whose shapes and sizes are same

Let ABC and DEF be two triangles in which AB = DE, BC = EF, AC = DF and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F respectively. Then, â–³ABC  â–³DEF

“CPCT” means corresponding parts of congruent triangles.

CRITERIA FOR CONGRUENCE OF TRIANGLES:

Side-angle-Side:

I      SAS) Congruence rule: Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding sides and the included angle of the other triangle.


If inâ–³sABC and DEF, AB = DE, AC = DF and ∠BAC =∠EDF

Then,           â–³ABC  â–³DEF

It is called SAS congruence rule i.e. side-angle-side]

II     Angle-Side-Angle (ASA) Congruence rule: Two triangles are congruent if two angles and the included side of one triangle are equal to two corresponding angles and the included side of other triangle.


CONGRUENCE RULE:

i.e. Angle-Side-Angle(ASA) congruence rule may be called Angle-Angle-Side (AAS) congruence rule.

III     Side-Side-Side(SSS) congruence rule:

If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.


If in â–³sABC and DEF, AB = DE, BC = EF and AC = DF

Then, â–³ABC  â–³DEF                               

[It is called SSS congruence rule i.e. side-side-side]

IV    Right angle-Hypotenuse-Side (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.

If ABC and DEF are two right triangles in which

∠B = ∠E = 90o, AC = DF and AB = DE

Then, â–³ABC  â–³DEF 

Theorem:     Angles opposite to equal sides of an isosceles triangle are equal

Converse of Theorem: The sides opposite to equal angles of a triangle are equal:

In â–³ABC if ∠B = ∠C

Then, AB = AC

INEQUALITIES IN A TRIANGLE:

Theorem:  If two sides of a triangle are unequal, the angle opposite to the longer side is larger (or greater)

Let ABC be a triangle in which AC > AB and AC > BC.

Then, ∠B > ∠A and ∠B > ∠C

The side opposite to the largest angle is the longest.

Theorem:     The sum of any two sides of a triangle is greater than the third side.

Let ABC be a triangle and AB, BC and AC are its corresponding sides.

Then, AB + BC > AC

AB + AC > BC

and AC + BC > AB

4. QUADRILATERIALS

Quadrilateral is a closed figure with four sides:

Angles’ Sum property of a Quadrilateral:

Theorem:     The sum of the angles of a quadrilateral is 360o

Types of Quadrilaterals:

I      A Trapezium: In a quadrilateral if one pair of opposite sides is parallel, then it is called a trapezium (Fig.3) i.e. If AB || CD then quadrilateral ABCD is a trapezium.


II     A parallelogram: In a quadrilateral if both pairs of opposite sides are parallel and equal, then it is called a parallelogram (Fig.4) i.e., AB || CD and AB = CD; AD || BC and AD = BC, then ABCD is a parallelogram.

III     A Rectangle: In a quadrilaterals (parallelogram) if all angles are right angles, then it is called a rectangle (Fig.5) i.e. AB || CD, AB = CD, AD || BC; AD = BC and ÐA = ÐB = ÐC = ÐD = 90o, then ABCD is a rectangle.

IV    A Rhombus: In a quadrilaterals (parallelogram) if all sides are equal, then it is called a rhombus (Fig.6), i.e., AB || CD, AD || BC and AB = BC = CD = DA, then ABCD is a rhombus.

V     A Square: In a quadrilateral (parallelogram) if all sides are equal and all angles are 90o, then it is called a square (Fig.7) i.e. AB || CD, AD || BC, AB = BC = CD = DA and ∠A = ∠B = ∠C = ∠D = 90o

VI    A Kite: In a quadrilateral ABCD (Fig.8), if AD = CD and AB = CB, then it is called a kite; i.e., two pairs of adjacent sides are equal but it is not a parallelogram.

PROPERTIES OF A PARALLELOGRAM:

Theorem: A diagonal of a parallelogram divides it into two congruent triangles.

Theorem:  If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.

Theorem:  In a parallelogram, opposite angles are equal.

             

             Here, ∠A = ∠C and ∠B = ∠D

Theorem: If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.

Theorem: The diagonals of a parallelogram bisect each other.

Here, OA = OC and OB = OD

Converse of above Theorem: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Theorem: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel

Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side.

Converse of above Theorem: The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.

5. CIRCLES

The collection of all the points in a plane which are at a fixed distance from a fixed point n the plane, is called circle.

Centre of the Circle: The fixed point is called the centre of the circle O is the centre of the circle in Fig.1.

Radius of the circle: The fixed distance from the centre and circumference of the circle is called the radius of the circle. OA = OC = r is the radius of the circle. We can draw infinite radii in a circle and all are equal in length.

Chord of the circle: The line segment which joins two points on the circumference of a circle is known as the chord of the circle. The chord of a circle does not pass through the centre of the circle. CD is a chord of the circle in Fig.2.

Diameter of the circle: The chord, which passes through the centre of the circle, is called a diameter of the circle. We can drawn infinite diameters in a circle and all are equal in length. In Fig.2,  AOB is a diameter of the circle. It is denoted by d.


It is said that a diameter is the longest chord of a circle.
A circle divides the plane on which it lies into following three parts in Fig.3

(i)    Interior of the circle: The plane which exists inside of a circle or the region inside of a circle is known as the interior of the circle.

(ii)   Circle: The geometrical figure which is surrounded by a circular line segment or a circle is a collection of all those points in a plane that are at given constant distance from a given fixed point in the plane.

(iii)  Exterior of the circle: The plane which exists outside of a circle or the region out side of a circle is known as the exterior of the circle.

Arc of a circle: A continuous piece of a circle is called an arc of the circle.

Minor arc: The shorter (smaller) arc of a circle is called minor arc. In Fig.4, PQ is the minor arc.


Major arc: The longer arc of a circle is called major arc. In Fig.4; PRQ is the major arc in Fig.4.

Semi circle: If P and Q are ends of a diameter then both arcs are equal and each is called a semi circle, i.e., PXQ and PYQ are equal arcs having a semi-circle in Fig.5. It is also called semicircular region.

Circumference: The length of the complete circle is called the circumference of the circle. It is denoted by C in Fig.6,


i.e. Circumference of the circle (C) = 2πr ;  where π = 22/7 or 3.14

Semi Circumference: Half length of the complete circle is called the semi-circumference of the circle. Both semi-circumferences of the circle are equal in length in Fig.7,

i.e.  Semi circumference = π.r

Segment of the circle: The region between a chord and either of its arcs is called a segment of the circle.

Minor Segment: The smaller region between a chord and smaller arc is called the minor segment of the circle, i.e. PXQ is the minor segment of the circle in Fig.8

Major segment: The bigger region between a chord and bigger arc is called the major segment of the circle, i.e., PYQ is the major segment of the circle in Fig.8.

Minor sector: When a circle is divided by its two radii, the smaller region of the circle is called minor sector, e.g., OAXB is the minor sector of the circle in Fig.9.

Major Sector: When a circle is divided by its two radii, the bigger region of the circle is called major sector, e.g. OAYB is the major sector of the circle Fig.9.

Theorem: Chords of a circle subtend equal angles at the centre.


Converse Theorem: Prove that if the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

Theorem:  Prove that the perpendicular from the centre of a circle to a chord bisects the chord.

Converse Theorem: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

Theorem:     There is one and only one circle passing through three non-collinear points.

Theorem: The length of the perpendicular from a point to a line is the distance of the line from the point.


Out of these line segments, the perpendicular from P to AB i.e. PM will be the least. Hence, this least length PM has to be the distance of AB from P.

Theorem: Equal chords of a circle are equidistant from the centre.



Converse Theorem: Prove that chords equidistance from the centre of a circle are equal in length

ANGLES SUBTENDED BY AN ARC OF A CIRCLE:

Theorem: If two chords of a circle are equal, then their corresponding arcs are congruent and conversely, if two arcs are congruent, then their corresponding chords are equal.

Let AB and CD be two chords of a circle with centre O.

             Then, AXB = CYD

Converse:  If AXB = CYD in a circle with centre O, then chord AB = chord CD

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.

Here, ∠BOC = 2 × ∠BAC

Theorem: Angles in the same segment of a circle are equal

Here, ∠BAD = ∠BCD

Theorem: Angle in a semicircle is a right angle.

Here, ∠BAC = 90o

Theorem:  If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment; four points lie on a circle (i.e., they arc concyclic). In

Fig.21


      
Here, A, B, C, D are concylic.

Cyclic Quadrilateral: A quadrilateral is called cyclic if all the four vertices of it lie on a circle.

Theorem: The sum of either pair of opposite angles of cyclic quadrilateral is 180o.

Here, ∠A + ∠C = 180o and ∠B + ∠D = 180o.

Converse Theorem: If the sum of a pair of opposite angles of a quadrilateral is 180o, the quadrilateral is cyclic.

Secant:  A line which intersects a circle in two distinct points is called a secant of the circle, e.g. in figure PQ is the secant of a circle ABCD with centre O in figure.

Tangent: A tangent to a circle is a line that intersects the circle in exactly one point. i.e., PQ is a tangent of a circle ABCD with centre O. And the touching point (point of contact) of the tangent PQ be R in figure. We can also say that there is only one tangent at a point of the circle, i.e., the common point of the tangent and the circle is called the point of contact and the tangent is said to touch the circle at the common point.

Theorem:    The tangent at any point of a circle is perpendicular to the radius through the point of contact.




Important:

(i)       At any point on a circle there can be one and only one tangent.

(ii)      The line containing the radius through the point of contact is also called the ‘normal’ to the circle at the point.

Theorem:   The lengths of tangents drawn from an external point to a circle are equal.

Hence, if PA and PB are two tangents from a point P to a circle with centre O, then PA = PB

6. COORDINATE GEOMETRY

CO-ORDINATE GEOMETRY:

It is a branch of Mathematics in which geometric problems are solved through algebra by using the coordinate system. So it is known as coordinate geometry.

CARTESIAN SYSTEM:

X'X and Y'Y two number lines are taken such that X'X is horizontal and Y'Y is vertical and they are crossing each other at their zeroes or origins. The horizontal line X'X is called X-axis and the vertical line Y'Y is called Y-axis. The point on which X'X and Y'Y intersect each other is called origin and is denoted y O. The positive numbers lie on the directions OX and OY are called the positive directions of the x-axis and y-axis, respectively. Similarly, OX' and OY' are called the negative directions of the X-axis and the Y-axis, respectively.


The axes divide the plane into four parts and each part is called quadrant. In anticlockwise they are called quadrant-I quadrant-II, quadrant-III and quadrant-IV. Hence, the plane consists of the axes and these quadrants. So, this plane is called Cartesian plane, or the coordinate plane or the XY-plane. The axes are called the coordinate axes.

X-COORDINATE:

The X-coordinate of a point is its perpendicular distance from Y-axis measured along the X-axis. The X-coordinate is also calld the abscissa.

Y-COORDINATE:

The Y-coordinate of a point is its perpendicular distance from X-axis. The Y-coordinate  is also called ordinate.

Distance Formula:                        

The distance between two points’ p(x1, y1) and Q(x2, y2) is given by

SECTION FORMULA:

The coordinates of point P which divide the straight line joining two points (x1, y1) and (x2, y2) internally in the ratio m1 : m2 are,

Corollary: COORDINATES OF THE MID-POINT:

If P is the mid-point of AB, then it will divide AB in the ratio of 1 : 1, then coordinates of P are :


Area of a Triangle:

Area of â–³ABC with the given vertices (x1, y1) and (x2, y2):

7. AREAS OF PARALLELOGRAMS AND TRIANGLES

Theorem: Parallelograms on the same base and in between the same parallels are equal in area.

Here, ar(ABCD) = ar(ABEF)

Theorem:  Two triangles on the same base (or equal bases) and in between the same parallels are equal in area.


Here, ar(â–³ABC) = ar(â–³DBC)

Theorem: The area of a triangle is half the product of its base (or any side) and the corresponding altitude (or height).

Here, ar(â–³ABC) = 1/2 × base × height.

IMPORTANT FORMULAE:

1.    Area of a rectangle = Length × Breadth = l × b

2.    Area of a square = (side)2 = a2

3.    Area of a triangle = 1/2 × Base × Height = 1/2 × b × h

4.    Area of a triangle =

[Where s = a + b + c and a, b, c are the corresponding sides of the â–³]
                       2

5.    Area of a parallelogram = Base × Height = b × h

6.    Area of a rhombus = 1/2 × d1 × d2

[Where d1 and d2 are the lengths of the two diagonals of the rhombus]

7.    Area of a Trapezium = 1/2(a + b) × h

[Where a and b are the lengths of opposite parallel lines and h is the distance between the parallel lines]

Perimeter and Area of different types of triangles:

(1)   Right Angled Triangle

Let be a right angled triangle in which, then

(i)  Perimeter = AB + BC + AC
(ii) Area = 1/2 × Base × Height 
(iii) AC2 = AB2 + BC2     (Pythagoras Theorem)

(2)   Isosceles Triangle:

Let  be an isosceles triangle in which AB = AC = a and BC = x. Let  then,

(i)  Perimeter = AB + BC + AC = 2a + x
(ii) Area = 1/2 × Base × Height    

3.    Equilateral Triangle:


Let  be an equilateral triangle in which

AB = BC = AC = a

(i) Perimeter =  3a
(ii) Altitude =  
(iii) Area =  

HERON’S FORMULA

Heron was born in about 10AD in Alexandria in Egypt. His work on mathematical and physical subjects is so numerous and varied that he is considered to be an encyclopedia writer in these field. His geometrical work deals largely with problems on mensuration.

The formula given by Heron is a famous formula for calculating area of a triangle in terms of its three sides.

Let a, b and c are the sides of the triangle and s is semi perimeter i.e. s = a + b + c 
                                                                                                                                      2 


This formula can be used for any triangle to calculate its area and it is very useful where it not possible to find the height of the triangle easily.

SOME IMPORTANT FORMULAS RELATED TO PLANE FIGURES

(1)   Rectangle

Let ABCD be a rectangle with length l and breadth b, then

(i) Perimeter =  2 (l + b)
(ii) Area = l × b    
(iii) Diagonal =

(2)   Square

Let ABCD be a square with each side equal to a, then

(i) Perimeter =  4a
(ii) Area = a2 
(iii) Diagonal =
(iv)  Area  = (diagonal)2
(v) Side of square =

(3)   Parallelogram

Let a parallelogram ABCD with adjacent sides a and b with diagonal d.

Let  and DE = h, then

(i) Perimeter =  2 (sum of adjacent sides) = 2 (a + b)
(ii) Area = Base × Height  = a × h
(iii)   If         s   = a + b + c         ,  then
    2 
   
  Area of parallelogram   =    

(4)   Rhombus

Let ABCD be a rhombus with each side equal to a. Let d1 and d2 are diagonals, then

(i) Area = 1/2 x d1 x d2
(ii) Perimeter =
(iii) Each side  =

(5)   Trapezium

Let ABCD be a trapezium in which AB || DC such that

AB = a and CD = b  then,

Area of trapezium   =   1/2 × (sum of parallel sides) × (distance between them)   

=   1/2 (a + b) x h

(6)   Quadrilaterals

(i)  Let ABCD be a quadrilateral in which length of diagonals = AC = d

Let DE | AC and BF | ACsuch that DE = h1 and BF = h2

So, area of quadrilateral  =  1/2 x d x (h1 + h2)                             

(ii)  Let ABCD be a kite then diagonals AC and BE are mutually perpendicular.

Let AC = d1 and BD = d2.

Area of kite  =     (product of the diagonals)                 

=    1/2 x d1 x d2      

(iii)     Let ABCD be a cyclic quadrilateral with sides a, b, c and d,


Then, area of cyclic quadrilateral

=   

Where      s     =    a + b + c
                                  2           

8. SURFACE AREAS AND VOLUMES

SOLID

A physical body which occupies some space is called a solid. It has three dimensions in space called length, breadth and height.

Example: a pen, a match box, a room etc.

VOLUME

The space occupied by a solid body is called its volume. Cubic centimeters (cm3) and cubic meters (m3) are the common units of volume.

Cuboid or Rectangular Parallelepiped: It is a solid with six rectangular faces.

If length, breadth and height of a cuboid are l, b and h respectively then,

(i)       Volume of cuboid       =   l × b × h

                                        =    area of base × height cu. units

(ii)      Total surface area of

cuboid                 =    2 (lb + bh + hl) sq. units

(iii)     Curved surface area of cuboid or surface

Area of 4 walls     =    2 (l + b) × h sq. units

(iv)     Diagonal of cuboid     =     units

(v)      Height of cuboid  =      volume     units    
                                                base area    

(vi)     Area of base    =       volume     sq. units   
                                               height           

(vii)    Surface area of cuboid, in which top face is open = lb + 2 (bh + hl) sq. units

(viii)   Diagonals of faces of cuboid  =      units

(ix)     Perimeter of the cuboid   =   4(l + b + h) units

CUBE

Cube is a rectangular solid. It has six faces in which every face is a square. Let each edge of a cube measures ‘a’ then

(i)       Volume of a cube = a3 cu. units

(ii)      Total surface area of cube = 6a2 sq. units

(iii)     Curved surface area of cube = 4a2 sq. units

(iv)     Diagonal of cube = a√3  units

(v)      Edge of cube = (volume)1/3 units

(vi)     Diagonal of face of the cube =  a√2 units

(vii)    Perimeter of the cube = 12 a units

RIGHT CIRCULAR CYLINDER

A solid generated by the revolution of a rectangle about one of its sides is called a right circular cylinder. In given figure OA is called the axis of the cylinder.

Let OA = height of the cylinder = h and AB = OC = radius of the base of the cylinder = r

then we have                                                                    
(i)  Area of the base of a right circular cylinder =  πr2 sq. units

(ii)  Area of the curved surface of a right circular cylinder

=    Circumference of the base × Height

=    2πrh  sq. units

(iii)  Total surface area of a right circular cylinder

=    Area of curved surface + 2 × Base area

=    (2πrh + 2πr2) sq. units

=    2πr (r + h)sq. units

(iv)  Volume of a right circular cylinder =    Area of base × height

=   (πr2) x h cu. units

=    πr2h cu. units

HOLLOW CYLINDER

A hollow cylinder is a solid bounded by two coaxial cylinders of the same height and different radii. If R and r be the external and internal radii of a hollow cylinder and h be its height, then

(i)  Each base surface area   =  π(R2 - r2) sq. units

(ii)  Curved surface area  =   (External surface area) + (Internal surface area)

=    2πRh + 2πrh

=    2πh (R + r)  

(iii)     Total surface area  =   (External surface area) + (Internal surface area) + 2 (base area)

=    2πRh + 2πrh + 2π(R2 - r2)

=    2πh(R + r) +  2π (R + r) (R - r)

=     2π(R + r)( h + R - r) sq. Units

(iv)     Volume of the material =   Exterior volume – interior volume

=   πR2h - πr2h

=   πh (R2 - r2)


RIGHT CIRCULAR CONE

If a right angled triangle is revolved about one of the sides containing a right angle, the solid thus formed is called a right circular cone.

A right circular cone can be also defined as a solid generated by revolving a line segment which passes through a fixed point and which makes a constant angle with a fixed line. In given figure we have, the fixed point A is called vertex of the cone and the fixed line AO is called the axis of the cone.

The base of a right circular cone is in circular shape such that the line joining vertex to the center of the circle is perpendicular to the base.

The length of the line segment joining the vertex to any point on the circular edge of the base is called the slant height of the cone.

Let Height of the cone   =  OA = h

Radius of the base of the cone = OB = r

And slant height of the cone =  OC = l

Then we have
(i)       Slant height   =    l      =     units

(ii)      Area of base    =    πr2h sq. units

(iii)     Volume    =    1/3 × (Area of base) × Height

=    1/3 πr2h cu units

(iv)     Curved surface area    = πrl sq. units

(v)      Total surface area      =   Area of circular base + curved surface area

=  (πr2 + πrl )  sq. units

=    πr (r + l) sq. units

Note: (i) If the base of a cone is not circular or if the line joining the vertex to the centre of the base is not perpendicular to the base then the cone is not right circular cone.

(ii) If a circle is revolved about its one of the diameter, then the solid formed is a sphere. A sphere can be described as a set of all those points in space, which are equidistant from a fixed point.

The fixed point is called the centre of the sphere and the constant distance between the centre and any point on the sphere is called radius of the sphere.


Let the radius of the sphere be r, then

(i)       Volume of the sphere      =  4/3 πr3 cu. units    

(ii)      Surface area of the sphere      =  4πr2 sq.units

HEMISPHERE

A plane passing through the centre of a sphere divides the sphere into two equal parts. Each part is called a hemisphere.


Let the radius of the hemisphere be r, then

(i)       Volume of the hemisphere = 2/3 πr3 cu. units

(ii)      Curved surface area = 2πr2 sq. units             

(iii)      Total surface area = 3πr2 sq. units

SPHERICAL SHELL

The difference of two solid concentric spheres is called a spherical shell.


Let the outer radius and inner radius of a spherical shell are R and r respectively, then

(i)       Volume of spherical shell =  4/3π (R3 - r3cub. Units

(ii)      External surface area =  4πRsq. units         

(iii)     Internal surface area =  4πr2sq. units

SURACE AREA OF COMBINATION OF SOLIDS:

I.     Total surface area of the solid given in figure = Curved surface area of one hemisphere


+ Curved surface area of right circular cylinder
+ Curved surface area of other hemisphere

= 2πr2 + 2πrh + 2πr2

= 4πr2 + 2πrh

= 2πr (2r + h)

II.    Total surface area of the solid [like toy / top (lattu)] (given in figure)

= Curved surface area of hemisphere + curved surface area of cone

= 2πr2 + πrl

= πr (2r + l)

FRUSTUM OF A CONE:

Let ‘h’ be the height, l the slant height and r1 and r2 the radii of the ends (r1 > r2) of the frustum of a cone in figure, then


(i)   Volume of the frustum of the cone =

(ii)  Curved surface area of the frustum of the cone = π(r1 + r2)

Where l =  

(iii) Total surface area of the frustum of the cone =       

Where l =  

9. TRIGONOMETRY

Trigonometry is a branch of mathematics and by using some mathematical techniques, we can find the distances or heights. The word “Trigonometry” is derived from the Greek words “tri” (means three), ‘gon’ (means sides) and ‘metron’ (measure). Actually, Trigonometry is the study of relationships between the sides and angles of a triangle. Trigonometry is one of the most ancient subjects studied by scholars all over the world. The astronomers used trigonometry to calculate distance from the Earth to the planets and stars. Trigonometry is also used in geography to construct maps, determine the position of an island in relation to the longitudes and latitudes, etc.

Trigonometric Ratios :           

Let ABC be a right triangle. In figure, ∠CAB is an acute angle. BC(p) is the side opposite to ∠A, AB (b) is the side adjacent to ∠A and AC is the hypotenuse.


Similarly, in figure, ∠ACB = ∠C is an acute angle. BC (b) is the side adjacent to ∠C, AB (p) is the side opposite to ∠C and AC is the hypotenuse (h), ‘p’ is perpendicular and ‘b’ is the base.

Now, The Trigonometric Ratios of ∠A in right triangle ABC (in figure.) are as given below:




 


Hence, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides.

In fact, the ratios CosecA, SecA and CotA are the reciprocals of the ratios sinA, cosA and tanA.

i.e.,       sinA = 1/cosecA ; cosecA = 1/sinA

cosA = 1/secA ; secA = 1/cosA

tanA = 1/cotA ; cotA = 1/tanA

Also,

tanA = sinA/cosA ; cotA = cosA/sinA

The values of the trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle, if the angle remains the same.

All six trigonometric ratios of an acute angle can be represented by θ (theta), ß (Bita), Y (Gama), π (pie),  ψ (Sie),  λ (Lamda), δ (delta) etc.

Trigonometric Ratios of Some Specific Angles:

Values of Trigonometric ratios of 0o to 90o : (Table) :

Some Basic Trigonometric Identities:

(1)  sin2A + cos2A = 1
       
(2) 1 + tan2A = sec2A
       
(3) 1+ cot2A = cosec2A
       
(4)  sin(90o - A)  = cosA
       
(5) cos(90o - A) = sinA
       
(6) tan(90o - A) = cotA
       
(7)   cot(90o - A) = tanA
       
(8) sec(90o - A) = cosecA
       
(9) cosec(90o - A) = secA

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