JMATHSLEARNING

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.

noreply@blogger.com (Euclid_mathematics_classes) — Tue, 04 Jul 2017 14:55:00 +0000

Given : A ▲ABC whose side BC has been produced to D forming exterior angle ∠4
To Prove : ∠4 = ∠1 + ∠3
Proof : We have, ∠1 + ∠2 + ∠3 = 180°
( Sum of the angles of a triangle)
Also, ∠2 + ∠4 = 180°
( Linear pair)
∴ ∠2 + ∠4 = ∠1 + ∠2 + ∠3
Hence, ∠4 = ∠1 + ∠3.

Theorem : The sum of the angles of a triangle is 180°

noreply@blogger.com (Euclid_mathematics_classes) — Sat, 24 Jun 2017 12:29:00 +0000

Given A △ABC
To Prove ∠1 + ∠2 + ∠3 = 180°
Construction Through A, draw a line DE parallel to BC.
Proof DE ॥ BC and AB is the transversal.
∴ ∠1 = ∠4 (Alt. Int. angle) ............ (i)
Again, DE ॥ BC and AC is the transversal.
∴ ∠2 = ∠5 (Alt. Int. angle) ............ (ii)
On adding the corresponding sides of (i) and (ii)
∴ ∠1 + ∠2 = ∠4 + ∠5
On adding ∠3 on both sides
So, ∠1 + ∠2 + ∠3 = ∠4 + ∠5 + ∠3
But, ∠4 + ∠5 + ∠3 = ∠ DAE = 180°
( ∵ ∠DAE is a straight angle)
∴ ∠1 + ∠2 + ∠3 = 180°
Hence, the sum of the angles of a triangle is 180° .

Median, Altitude, Incentre and Circumcircle

noreply@blogger.com (Euclid_mathematics_classes) — Fri, 23 Jun 2017 02:25:00 +0000

Median

The median of a triangle corresponding to any side is the segment joining the midpoint of that side with the opposite vertex.

In the given figure, D, E, F are the respective midpoints of sides BC, CA and AB of triangle ABC.

∴ AD is the median, corresponding to side BC. BE is the median, corresponding to side CA. CF is the median, corresponding to side AB.

The medians of a triangle are concurrent, i.e., they interest each other at the same point.

Cendroid: The point of intersection of all the three medians of a triangle is called its centroid.

In the above figure, the medians AD, BE and CF of triangle ABC interest at the point G.

∴ G is the centroid of triangle ABC.

Altitudes

The altitude of a triangle corresponding to any side is the length of perpendicular drawn from the opposite vertex to that side.

The side on which the perpendicular is being drawn, is called its base.

In the given figure, AL ⟘ BC; BM ⟘ CA and CN ⟘ AB.

∴ AL is the altitude, corresponding to base BC.

BM is the altitude, corresponding to base CA.

CN is the altitude, corresponding to base AB.

A triangle has three altitudes.

The altitudes of a triangle are concurrent.

Orthocentre: The point of intersection of all the three altitudes of a triangle is called its orthocentre.

In the above figure, the three altitudes AL, BM and CN of △ ABC interest at a point H.

∴ H is the orthocentre of △ ABC.

Incentre of a Triangle

The point of intersection of the internal bisector of the angles of a triangle is called its incentre.

In the given figure, the internal bisectors of the angles of △ ABC intersect at I.

∴ I is the incentre of △ ABC, Let ID ⊥ BC.

Then, a circle with centre I and radius ID is called the incircle of △ ABC.

Circumference of a Triangle

The point of intersection of the perpendicular bisectors of the sides of a triangle is called its circumference.

In the given figure, the right bisectors of the sides of △ ABC intersect at O.

∴ O is the circumference of △ ABC.

With O as centre and radius equal to OA = OB = OC, we draw a circle passing through the vertices of the given triangle.

This circle is called the circumference of △ ABC.

Triangle definition | Equilateral Isosceles Scalene and Acute angled Right angle Obtuse angled Triangle

noreply@blogger.com (Euclid_mathematics_classes) — Wed, 21 Jun 2017 02:26:00 +0000

Triangle

A plane figure bounded by three line segments is called triangle.

We denote a triangle by the symbol△ .

A △ABC has:

(i) Three vertices, namely A, B and C.

(ii) Three sides, namely AB, BC and CA.

(iii) Three angles, namely ∠A, ∠B and ∠C.

A triangle has six parts— three sides and three angles

Type of Triangles:

1. On the basis of sides:

(i). Equilateral Triangle: A triangle having all sides equal is called an equilateral triangle.

In the given figure, ABC is a triangle in which AB = BC = CA.

∴ △ABC is an equilateral triangle.

(ii). Isosceles Triangle: A triangle having two sides equal is called an isosceles triangle.

In the given figure, ABC is a triangle in which AB = AC.

∴ △ABC is an isosceles triangle.

(iii). Scalene Triangle: A triangle in which all the sides are of different length is called a scalene triangle.

In the given figure, PQR is a triangle. In which PQ ≠ QR ≠ PR.

∴ △PQR is a scalene triangle.

2. On the basis of angles:

(i). Acute-angled Triangle: A triangle in which every angle measures more than 0⁰ but less than 90⁰ is called an acute angled triangle.

In the given figure, ABC is a triangle in which every angle is an acute angle.

∴ △ABC is an acute-angled triangle.

(ii). Right-angled triangle: A triangle in which one of the angles measures 90⁰ is called a right-angled triangle or simply a right triangle.

In a right-angled triangle, the side opposite to the right angle is called its hypotenuse and the remaining two sides are called its legs.

In the given figure, △ABC is a right triangle in which , AC is the hypotenuse, and AB, BC are its legs.

(iii). Obtuse-angled triangle: A triangle in which one of the angles measures more than 90⁰ but less than 180⁰ is called an obtuse-angled triangle.

In the given figure, PQR is a triangle in which ∠PQR is an obtuse angle.

∴ △PQR is an obtuse-angled triangle.

Perimeter of a triangle: The sum of the lengths of three sides of a triangle is called its perimeter.

What is angle | Type of angles

noreply@blogger.com (Euclid_mathematics_classes) — Mon, 19 Jun 2017 01:00:00 +0000

Angle

Two rays starting from a common point form an angle. In ∠ AOB, O is the vertex, OA and OB are the two rays.

−→.

Measure of a angle

The amount of turning from OA to OB is called the measure of ∠ AOB, written as m∠ AOB. An angle is measured is degrees denoted by ‘°’.

Types of angles

(i) Acute angle: An angle whose measure is greater than 0° but less than 90° is called an acute angle.

Example: 15° ,30° ,60° ,75° ,etc.

∠ AOB = 30° is an acute angle.

(ii) Right angle: An angle whose measure is 90° is called a right angle.
∠AOB = 90°is a right angle.

(iii) Obtuse angle: An angle whose measure is greater than 90° and less than 180° is called an obtuse angle.

Example: 100°, 110°, 120°, 140°, etc.

∠AOB = 110° is an obtuse angle.

(iv) Straight angle: When the rays of an angle are opposite rays forming a straight line, the angle thus formed is a straight angle and its measure is 180°.

∠AOB= 180° is a straight angle.

(v) Reflex angle: An angle whose measure is more than 180° but less than 360° is called a reflex angle.

∠AOB= 220° is a reflex angle.

(vi) Complete angle: The angle formed by OP and OQ is one complete circle, that is 360°.Such an angle is called a complete angle.

This complete rotation is divided into 360 equal parts. Each part measures 1°.

1° = 60 minutes, written as 60’.

1’ = 60 seconds, written as 60”.

Bisector of an angle

A ray AC is called the bisector of ∠BAD, if m∠BAC = m∠DAC.

In this case ∠BAC =∠DAC = 1/2∠BAD .

Related Angles

(i) Complementary angles: If the sum of the measures of two angle is 90°, then the two angles are called complementary angles. Here each angle is the complement of the other. The complement of 30° is 60°and the complement of 60° is 30°.

(ii) Supplementary angles: If the sum of the measures of two angle is 180°, then the two angles are called supplementary angles. Here each angle is the supplement of the other.

The supplementary angle of 120° is 60° and the supplementary angle of 60° is 120°.

Adjacent angles

If two angles have the same vertex and a common ray, then the angles are called adjacent angles. ∠BAC and ∠CAD are adjacent angles (i.e . ∠x and ∠y) as they have a common ray AC, a common vertex A and both the angle ∠BAC and ∠CAD are on either side of the common ray AC, a common vertex A and both the angle ∠BAC and ∠CAD are on either side of the common ray AC.

Linear Pair of Angle

Two adjacent angles are said to form a linear pair of angles, if their non-common ray are two opposite rays.

In the adjoining figure, ∠ AOC and ∠ BOC are two adjacent angles whose non-common rays OA and OB are two opposite rays, i.e., BOA is a line.

∴ ∠ AOC and ∠ BOC form a linear pair of angles.

Angle at a point

Four angles are formed at the point ‘O’. The sum of the four angles formed is 360°. (i.e) ∠1 + ∠ 2 + ∠ 3 + ∠ 4 = 360°

Vertically opposite angles

If two straight lines AB and CD intersect at a point ‘O’. Then ∠ AOC and ∠ BOD form one pair of vertically opposite angles and ∠ DOA and ∠COB form another pair of vertically opposite angles.

Parallel lines

Two lines l and m in a plane are said to be parallel, if they have no point in common and we represent l∥ m

The distance between two parallel lines always remains the same.

Transversal

A straight line which cuts two or more straight lines at distinct points is called a transversal.

The Angles formed when a transversal cuts two lines

Let AB and CD be two lines, cut by a transversal t. Then, the following angles are formed.

(i) Pairs of corresponding angles:

(a) ∠ 1 = ∠5

(b) ∠ 4 = ∠ 8

(c) ∠ 2 = ∠ 6

(d) ∠ 3 = ∠ 7

(ii) Pair of alternate interior angles:

(a) ∠ 3 = ∠ 5

(b) ∠ 4 = ∠ 6

(iii) Pair of consecutive interior angles (Co-interior angle)

(a) ∠4 + ∠5 = 180°

(b) ∠ 3 +∠ 6 = 180°

Various Type of Numbers Chart

noreply@blogger.com (Euclid_mathematics_classes) — Fri, 16 Jun 2017 08:34:00 +0000

Definition | Point, Ray, Line, Collinear Points, Plane, Parallel Lines

noreply@blogger.com (Euclid_mathematics_classes) — Fri, 16 Jun 2017 01:21:00 +0000

Point

A point is an exact location.

A fine dot represents a point.

It is denoted by capital letter – A, B, P, Q, etc.

In the given figure, P is a point.

Line Segment

The straight path between two points A and B is called the line segment AB

The points A and B are called the end points of the line segment AB .
A line segment has a definite length.

Ray

A portion of a line which starts at a point and goes off in a particular direction to infinity is known as ray.

Ray has one end point A. A Ray has no definite length. We write a ray

Line

A line segment when extended indefinitely in both the directions is called the line .

A line has no end points. A line has no definite length. It is denoted by small letters l, m, n, etc.

Incidence Axioms on lines

A line contains infinitely many points.
Through a given point, infinitely many lines can be drawn.
One and only one line can be drawn to pass through two given points A and B.

Collinear Points

Three or more than three points are said to be collinear, if there is a line which contains them all.

Collinear Points

In the given figure A, B, C are collinear points, while P, Q, R are noncollinear.

Non collinear Points

Intersecting lines

Two lines having a common point are called intersecting lines.

The point common to two given lines is called their point of intersection.

In the given figure, the lines AB and CD intersect at a point O.

Concurrent Lines

Three or more lines intersecting at the same point are said to be concurrent.

In the given figure, lines l, m, n pass through the same point P and therefore, they are concurrent.

Plane

A plane is a surface such that every point of the line joining any two points on it, lies on it.

Examples: The surface of a smooth wall; the surface of the top of the table; etc.

Parallel Lines

Two lines l and m in a plane are said to be parallel, if they have no point in common and we represent l∥ m

The distance between two parallel lines always remains the same.

Euclid's Axioms & Postulates | examples

noreply@blogger.com (Euclid_mathematics_classes) — Wed, 14 Jun 2017 12:31:00 +0000

History

Thales
(640 BC – 546 BC)

The word ‘Geometry’ is derived from the Greek words ‘Geo’ means ‘Earth’ and ‘Metrein’ means to ‘ To Measure’. A Greek mathematician, Thales is credited with giving the first known proof. This proof was of the statement that a circle is bisected (i.e., cut into two equal parts) by its diameter. One of Thales’ most famous pupils was Pythagoras (572 BC), whom you have heard about. Pythagoras and his group discovered many geometric properties and developed the theory of geometry to a great extent. This process continued till 300 BC. At that time Euclid, a teacher of mathematics at Alexandria in Egypt, collected all the known work and arranged it in his famous treatise,

Euclid
(325 BC – 265 BC)

called ‘Elements’. He divided the
‘Elements’ into thirteen chapters, each called a book. These books influenced the whole world’s understanding of geometry for generations to come. Euclid listed 23 definition in book 1 of the ‘Elements’.

A Few the Euclid’s Definitions :

1. A point is that which has no part.

2. A line is breadthless length.

3. The ends of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

5. A surface is that which has length and breadth only.

6. The edges of a surface are lines.

7. A plane surface is a surface which lies evenly with the straight lines on itself.

Basic Geometrical Concepts:

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

Examples: 1. Doubles of equals are equal.
2. The whole is greater than each of its parts
3. A line contains infinitely many points.

Postulate: The basic facts related to geometry which are taken for granted, without proof are called postulate.

Examples: 1. A terminated line can be produced indefinitely.

2. All right angles are equal to one another.

Statements: A sentence which can be judged to be true or false is called a statement.

Examples: 1. The sum of the angles of a triangle is 180⁰, is a true statement.

2. The sum of the angles of a quadrilateral is 180⁰, is a false statement.

3. x + 10 > 12 is a sentence but not a statement.

Theorem: A statement that requires a proof, is called a theorem. Establishing the truth of a theorem is known as proving the theorem.

Examples: 1. The sum of the angles of a triangle is 180⁰.

2. The sum of all the angles around a point is 360⁰.

Corollary: A statement whose truth can easily be deduced from a theorem, is called its corollary.

Euclid’s Axioms:

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

Example: If a = b and b = c, then a = c

(2) If equals are added to equals, the wholes are equal.

Examples: If a = b and c = d

On adding

Then, a + c = b + d

(3) If equals are subtracted from equals, the remainders are equal.

Examples: If x = y and a = b

On subtracting

Then, x – a = y – b

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

Example: X coincide with X, hence X = X

(5) The whole is greater than the part.

Example: Suppose earth is the part of whole universe but universe is not a part of earth.

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

Example: If 2a = 2b

Then a = b

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

Example: If ½ x = ½ y

Then x = y

Euclid’s Five Postulates:

Postulate 1 : A straight line may be drawn from any one point to any other point.

Postulate 2 : A terminated line can be produced indefinitely.

Postulate 3 : A circle can be drawn with any centre and any radius.

Postulate 4 : All right angles are equal to one another.

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.

Algebraic Identites

noreply@blogger.com (Euclid_mathematics_classes) — Mon, 12 Jun 2017 14:17:00 +0000

Formulae for binomial

1. (a + b)² = a² + 2ab + b²

Proof : Taking L.H.S

(a + b)²= (a + b) . (a + b)

= a (a + b ) + b (a + b)

= a² + ab + ab + b²

(a + b)² = a² + 2ab + b² …….. (i)

2. (a – b)² = a² – 2ab + b²

Proof : Putting b = – b in equⁿ (i)

(a – b)² = a² + 2a(–b) + (– b)²

(a – b)² = a² – 2ab + b²

3. a² – b² = (a + b) . (a – b)

Proof : Taking R.H.S

(a + b) . (a – b) = a (a – b) + b (a – b)

= a² – ab + ab – b²

= a² – b²

4. (a + b)³ = a³ + b³ + 3ab (a + b) = a³ + b³+ 3a²b + 3ab²

Proof : Taking L.H.S

(a + b)³= (a + b) . (a + b)²

= (a + b) . (a² + 2ab + b²)

= a (a² + 2ab + b²) + b (a² + 2ab + b²)

= a³ + 2a²b + ab² + a²b + 2ab² + b³

= a³ + 3a²b + 3ab² + b³

(a + b)³= a³ + b³ + 3a²b + 3ab²= a³ + b³ + 3ab (a + b)..(ii)

5. (a – b)³ = a³ – b³ – 3ab (a – b) = a³ – b³– 3a²b + 3ab²

Proof : Putting b = – b in equation (ii)

(a – b)³ = a³ +(– b)³ + 3a(–b) [a +(– b)]

(a – b)³= a³ – b³ – 3ab (a – b) = a³ – b³– 3a²b + 3ab².....(iii)

6. (a³ + b³) = (a + b)(a² – ab + b²)

Proof : From (ii)

(a + b)³ = a³ + b³ + 3ab (a + b)

(a + b)³ – 3ab (a + b)= a³ + b³

a³ + b³ = (a + b)³ – 3ab (a + b)

= (a + b) [(a + b)² – 3ab]

= (a + b) [ a² + 2ab + b² – 3ab]

(a³ + b³) = (a + b) ( a² – ab + b²)

7. (a³ – b³) = (a – b)(a² + ab + b²)

Proof : From (iii)

(a – b)³ = a³ – b³ – 3ab (a – b)

(a – b)³ + 3ab (a – b)= a³ – b³

a³ – b³ = (a – b)³ + 3ab (a – b)

= (a – b) [(a – b)² + 3ab]

= (a – b) [ a² – 2ab + b² + 3ab]

(a³ – b³) = (a – b) ( a² + ab + b²)

formula for Trinomial

8. (a + b + c)² = a²+ b² + c² + 2ab + 2bc + 2ca

Proof : Taking L.H.S

(a + b + c)² = (a + b)² + c² + 2 (a + b) . c

= a²+ 2ab + b² + c² + 2ac + 2bc

(a + b + c)²= a² + b² + c² + 2ab + 2bc + 2ca

9. (a³ + b³ + c³ – 3abc) = (a + b + c)(a²+ b² + c² – ab – bc – ca)

Proof: Taking L.H.S

(a³ + b³+ c³ – 3abc) = (a + b)³ – 3ab (a + b) + c³ – 3abc

= (x³ + c³) – 3ab (x + c), where (a + b)= x

= (x + c)(x² – xc + c²) – 3ab (x + c)

= (x + c)[(x² – xc + c²) – 3ab]

= (a + b + c)[(a + b)² – (a + b)c + c² – 3ab]

= (a + b + c)[a² + 2ab + b² – ac – bc + c² – 3ab]

(a³ + b³+ c³ – 3abc) = ( a + b + c)(a²+ b² + c² – ab – bc – ca)

10. If (a + b + c) = 0 ⇒ a³+ b³+ c³= 3abc

Proof : We have

(a + b + c) = 0

(a + b) = – c

Cubing both sides

(a + b)³ = (– c)³

a³ + b³ + 3ab (a + b) = – c³

a³+ b³+ 3ab (– c) = – c³

a³+ b³– 3abc = – c³

a³+ b³+ c³ = 3abc

Remainder Theorem | Factor Theorem

noreply@blogger.com (Euclid_mathematics_classes) — Sun, 11 Jun 2017 15:01:00 +0000

Remainder Theorem

Let p(x) be a polynomial of degree ≥ 1 and let a be any real number. When p(x) is divided by ( x – a), then the remainder is p(a).

Proof : Suppose that when p(x) is divided by (x – a), the quotient is q(x) and the remainder is r.

∴ Dividend = Divisor x Quotient + Remainder

p(x) = (x – a) . q(x) + r ……….. (i)

Putting x = a in (i),

We get r = p(a)

Thus, when p(x) is divided by (x – a), then the remainder is p(a).

Example: Find the remainder when the polynomial

p(x) = x³ + 3x² – 2x +1 is divided by (x – 4).

Solution: p(x) = x³ + 3x² – 2x +1

x – 4 = 0 ⇒ x = 4

By the remainder theorem, we that when p(x) is divided by (x – 4), the remainder is p(4).

Now, remainder = p(4)

= (4³ + 3 x 4² – 2 x 4 + 1)

= (64 + 3 x 16 – 8 +1 ) = (64 + 48 – 7) = 105

Hence, the required remainder is 105.

Factor Theorem

Let p(x) be a polynomial of degree n > 1 and a be any real number.

(i). If p(a) = 0 then (x – a) is a factor of p(x).

(ii). If (x – a ) is a factor of p(x) then p(a) = 0.

Proof: (i). p(a) = 0.

On dividing p(x) by (x – a ), let q(x) be the quotient.

By remainder theorem, when p(x) is divided by (x – a), then remainder is p(a).

Therefore p(x) = (x – a) . q(x) + p(a)

⇒ p(x) = (x – a) . q(x) { ∴ p(a) = 0 }

⇒ (x – a) is a factor of p(x).

(ii) Let (x – a) be a factor of p(x).

On dividing p(x) by (x – a), let q(x) be the quotient.

Then, p(x) = (x – a) . q(x)

⇒ p(a) = 0 {putting x = a}

Thus, (x – a) is a factor of p(x) imply f(a) = 0

Example: Show that (x – 2) is a factor of the polynomial

p(x) = x³ – 2x² – 3x + 6

Solution: (x – 2) = 0 ⇒ x = 2

By the factor theorem, (x – 2) will be a factor of p(x) if p(2) = 0

Now, p(x) = x³ – 2x² – 3x + 6

⇒ p(2) = 2³ – 2 x 2² – 3 x 2 + 6

= 8 – 2 x 4 – 6 + 6

= 8 – 8

= 0

⇒ (x – 2) is a factor of p(x)

Hence, (x – 2) is a factor of the given polynomial p(x).

What is Polynomial | Various kinds of Polynomials

noreply@blogger.com (Euclid_mathematics_classes) — Fri, 09 Jun 2017 14:02:00 +0000

Constant

A symbol having a fixed numerical value is called a constant

Examples: 3, -6, 5/6, 0.23, π, etc. are all constants.

Variable

A symbol which may be assigned different numerical values us known as a variable.

Example: We know that the area of a circle is given by the formula A = πr², where r is the radius of the circle.

Here, π is constant, while A and r are variables.

Algebraic Expression

A combination of constants and variables, connected by some or all of the operations +, −, ×, and ÷ is known an algebraic expression.

Examples: x +2, x²+y², x – 1/x, etc.

Terms of an Algebraic Expression

The several parts of an algebraic expression separated by + or – operations are called the terms of the expression.

Examples:

(i). 3 + 4x – 7xy² + 9xy is an algebraic expression containing four terms, namely, 3, 4x, 7xy² and 9xy

(ii). X³+ 3x²y – 5xy² + y³ – 6 is an algebraic expression containing five terms, namely, X³, 3x²y, 5xy², y³ and 6.

Polynomial

An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial.

Examples:

(i). x³+ 5x²- 2x + 3 is a polynomial in one variable x.

(ii). 2p⁴ + 5p³ – 2p² + p – 5 is a polynomial in one variable y.

(iii). 3 + x – x^3/2+ 2x³ is an expression but not a polynomial since it contains a term containing x^3/2, where 3/2 is not a non-negative integer.

Degree of Polynomial in one variable

In case of a polynomial in one variable, the highest power of the variable is called the degree of the polynomial.

Examples:

(i). 3x – 4 is a polynomial in x of degree 1.

(ii). 2m² + 6m – 5 is a polynomial in m of degree 2.

Degree of a Polynomial in two or more variables

In case of polynomials in more than one variable, the sum of the powers of the variables in each term is taken up and the highest sum so obtained is called the degree of the polynomial.

Examples:

(i). 2x³ + 3x³y – 4xy + 3x – 5 is a polynomial in x and y of degree 4.

(ii). 3m⁴n² + 2m²n² – 5m + 4n – 2 is a polynomial in m and n of degree 6

Various Type of Polynomials

1. Polynomials of Various Degrees

(i). linear polynomial

A polynomial of degree 1 is called a linear polynomial.

Examples: (a). 2x – 5 is a linear polynomial in x.

(b). 3p + 4 is a linear polynomial in p.

(c). x + 2y +5 is a linear polynomial in a x and y.

(ii). Quadratic Polynomial

A polynomial of degree 2 is called a quadratic polynomial.

Examples: (a). x²+ 5x – 2 is a quadratic polynomial in x.

(b). 2z² – 3z + 2 is a quadratic polynomial in z.

(c). ab + bc + ca is a quadratic polynomial in a, b and c.

(iii). Cubic Polynomial

A polynomial of degree 3 is called a cubic polynomial.

Examples: (a). 2x³ + 3x² – 5x + 2 is cubic polynomial in x.

(b). x²y – 2xy + 3 is cubic polynomial in x and y.

(iv). Biquadratic Polynomial

A polynomial of degree 4 is called a biquadratic polynomial.

Examples:

(a). x⁴ + 3x³ – 6x² + 5x + 2 is biquadratic polynomial in x.

(b). x³y + xy² + 2xy – 5 is biquadratic polynomial in x and y.

2. Number of terms in a Polynomial:

(i). Monomial

A polynomial containing one nonzero term is called a monomial.

Examples: 2, 3x, xy², 5x³y etc.

(ii). Binomial

A polynomial containing two nonzero terms is called a binomial.

Examples: (5x + 3), (x² + y²), (x – 3y) etc.

(iii). Trinomial

A polynomial containing three nonzero term is called a trinomial.

Examples: (x²+ 3x +2), (3y – 2xy + 3xy²) etc.

Constant polynomial

A polynomial containing one nonzero term, consisting of a constant is called a constant polynomial. The degree of nonzero of constant polynomial is zero.

Examples: 2, - 4, 2/3 etc.

Zero Polynomial

A polynomial containing one term, namely zero only, is called a zero polynomial. The degree of a zero polynomial is not defined.

Properties of Irrational Numbers

noreply@blogger.com (Euclid_mathematics_classes) — Thu, 08 Jun 2017 14:03:00 +0000

1. Irrational numbers satisfy the commutative, associative and distributive laws for addition and multiplication.

2. (i) Sum of two irrationals need not be an irrational.

Example: Each one of (2 + √3) and (3 - √3) is irrational.

But, (2 + √3) + (3 - √3) = 5, which is rational.

(ii). Difference of two irrationals need not be an irrational.

Example: Each one of ( 4 + √5) and ( 2 + √5 ) is irrational.

But, ( 4 +√5) - ( 2 + √5 ) = 2, which is rational.

(iii). Product of two irrationals need not be an irrational.

Example: √5 is irrational.

But, √5 x √5 = 5, which is rational.

(iv). Quotient of two irrationals need not be an irrational.

Example: Each one of 3√5 and √5 is irrational.

But, 3√5/√5 = 3, which is rational.

3 (i). Sum of a rational and an irrational is irrational.

Example: 2 is rational and √3 is irrational then,

(2 + √3) is a irrational number.

(ii). Difference of a rational and an irrational is irrational.

Example: 5 is rational and √7 is irrational then,

(5 - √7) is a irrational number.

(iii). Product of a rational and an irrational is irrational.

Example: 2 is rational and √3 is irrational then,

(2√3) is a irrational number.

(iv). Quotient of a rational and an irrational is irrational.

Example: 2 is rational and √3 is irrational then,

(2/√3) is a irrational number.

Real Numbers | Properties

noreply@blogger.com (Euclid_mathematics_classes) — Wed, 07 Jun 2017 14:24:00 +0000

Real Numbers

A number whose square is non-negative, is called a real number.

Example: All rational and irrational numbers are real numbers.

Addition Properties of real numbers:

1. Closure Property : The sum of two real numbers is always a real number.

Example : Two real numbers are 1 and √2, then (1 + √2) is a real number.

2. Commutative Property : a and b are two real numbers, then

a + b = b + a

Example : two real numbers are (2 + √3) and (2 – √3), then

(2 + √3) + (2 – √3) = 4

(2 – √3) + (2 + √3) = 4

Then left hand side = Right hand side

Hence Commutative Property is verified.

3. Associative Property : a, b and c are three real numbers, then

a + ( b + c ) = ( a + b ) + c

Example : Three real numbers are 2 , (5 + √7) and (5 - √7)

2 + [(5 + √7) + (5 - √7)] = 2 + 10 = 12

[2 + (5 + √7) ] + (5 - √7) = (7 + √7) + (5 - √7) = 12

Then left hand side = Right hand side

Hence Associative Property is verified.

4. Existence of Additive Identity : For any real number a, then

0 + a = a + 0 = a

0 is called the additive identity for real numbers.

Example : for a real number 5, then

0 + 5 = 5 + 0 = 5

5. Existence of Additive Inverse : For each real number a, there exist a real number ( - a) such that a + (- a) = (-a) + a = 0 then,

a and ( -a ) are called the additive inverse (or negative) of each other.

Example : For a real number (2 – √3), then

Additive inverse of 2 – √3 is – (2 – √3 ) = (√3 – 2)

Multiplication Properties of Real Numbers :

1. Closure Property : The product of two real numbers is always a real numbers.

Example : Two real numbers are 1 and √2, then

1 х √2 = √2 is a real number.

2. Commutative Property : a and b are two real numbers, then

a х b = b х a

Example : Two real numbers are (2 + √3) and (2 – √3), then

(2 + √3)(2 – √3) = 2² – ( √3 )² = 4 – 3 = 1

(2 – √3)(2 + √3) = 2² – ( √3 )² = 4 – 3 = 1

Then left hand side = Right hand side

Hence Commutative Property is verified.

3. Associative Property : a, b and c are three real numbers, then

a х ( b х c ) = ( a х b ) х c

Example : Three real numbers are 2 , (5 + √7) and (5 - √7),

2 х [(5 + √7) х (5 - √7)] = 2 х [5² – (√7)² ] = 2 х [25 – 7 ]

= 2 х 18 = 36

[2 х (5 + √7)] х (5 - √7) = [10 + 2√7 ] х (5 - √7)

= 50 – 10√7 + 10 √7 - 2 х 7 = 50 – 14 = 36

Then left hand side = Right hand side

Hence Associative Property is verified.

4. Existence of Multiplicative Identity : For any real number, a then,

1 . a = a . 1 = a

1 is the multiplicative identity for any numbers.

Example : for a real number 5, then

1. 5 = 5 . 1 = 5

5. Existence of Multiplicative Inverse : For each non-zero real number a, there exist a real number 1/a such that a . 1/a = 1/a . a = 1 then,

a and 1/a are called the multiplicative inverse (or reciprocal ) of each other.

Example : For a real number (2 – √3) then,

Multiplicative inverse of (2 – √3) is 1/ (2 – √3) = (2 + √3)

6. Distributive Properties of Multiplication over Addition : a, b and c three real numbers, then

a x ( b + c ) = a x b + a x c

Example : Three real numbers are 2 , (5 + √7) and (5 - √7) then,

2 x [(5 + √7) + (5 - √7)] = 2 x 10 = 20

2 x (5 + √7) + 2 x (5 - √7) = 10 + 2√7 + 10 – √7 = 10 + 10 = 20

Then left hand side = Right hand side

Hence Distributive Property is verified.

Rational Numbers & Irrational Numbers | Examples

noreply@blogger.com (Euclid_mathematics_classes) — Mon, 05 Jun 2017 04:24:00 +0000

Rational Numbers

The numbers of the form a/b, where a and b are integers and b ≠ 0 are known as rational numbers.

Example: 1/2, -4/5, 0, -1, 2 etc.

A number which can be expressed as a terminating decimal or repeating decimal, is called a rational numbers.

Example: 5/3 = 1.666666....... = 1.6
16/45 = 0.355555 ...... = 0.35 etc.

Important result

all natural numbers, whole numbers and integers are rational numbers.

We can understand above terms with a picture diagram.

Irrational Numbers

A number which can neither be expressed as a terminating decimal nor a repeating decimal, is called an irrational number.

Thus, non-terminating, non-repeating decimals are irrational numbers.

Examples:

Type 1. 0.2020020002....... , 0.4040040004...... etc. are a non-terminating and non-repeating decimal, therefore these are irrational number.

Type 2: If n is positive which is not a perfect square, then √n is irrational, e.g. √2, √3, √5, √6, √7, √8, √10, √11 etc.

Type 3: If n is positive which is not a perfect cube, then ³√n is irrational, e.g. ³√2, ³√3, ³√4, ³√5, ³√6, ³√7, ³√9, ³√10 etc.

Important Result

𝛑 is irrational, while 22/7 is rational number.

What is Decimal Numbers? Terminating and Non-terminating | Recurring and Non-recurring decimal

noreply@blogger.com (Euclid_mathematics_classes) — Sun, 04 Jun 2017 12:15:00 +0000

Decimal Number

Decimal numbers are numbers that are written according to their place value. The decimal point is used to separate whole part from number value less than one.A symbol is used " . ", this is called decimal point.

Example: 65.894 where 65 is a whole part and 894 is a fractional part.

Type of Decimals

1. Terminating Decimals

2. Non-terminating Decimals

1. Terminating Decimals

Every fraction a/b can be expressed as a decimal. If the decimal expression of a/b terminates i.e. comes to end, then the decimal so obtained is called a terminating decimal.

Example: 1/2 = 0.5

5/4 = 1.25

13/5= 2.6

Thus, each of the numbers 1/2, 5/4 and 13/5 can be expressed in the form of a terminating decimal.

2. Non-terminating Decimals

While expressing a fraction in the form of decimal, when we perform division we get some remainder. If the division process does not end i.e. remainder is not equal to zero, then such decimal is known as non-terminating decimal.

Example: 5/3 = 1.666666....... = 1.6

16/45 = 0.355555 ...... = 0.35

These are non-terminating Decimals

Type of Non-terminating Decimals

1. Recurring Decimal

2. Non-recurring Decimal

We can understand above these terms with some examples.

Properties of Integers with examples

noreply@blogger.com (Euclid_mathematics_classes) — Sat, 03 Jun 2017 10:44:00 +0000

Properties of Integers

Addition Properties of Integers:

1. Closure Property:  The sum of two Integers is always an Integers.

Example: Let two Integers are -4 and -6 then

(-4) + (-6) = (-10) is an Integer.

2. Commutative Property: Let a and b are two Integers, then

a + b = b + a

Example: Let two integers are -5 and 6 then

(-5) + 6 = 1

6 + (-5) = 1

then (-5) + 6 = 6 + (-5)

Hence commutative property is verified

3. Associative Property: Let a, b and c are three integers, then
a + ( b + c) = (a + b ) + c
Example: Let three integers are 3, 5 and (-6) then
3 + { 5 + (-6) } = 3 + {5 - 6 } = 3 + (-1) = 2
{3 + 5 } + (-6) = 8 - 6 = 2
then 3 + { 5 + (-6) } =  { 3 + 5 }+ (-6)
Hence Associative property is verified

4. Existence of Additive Identity:  For any integers a , we have

a + 0 = 0 +a = a
0 is called the additive identity.
Example: 4 is any integer,
4 + 0 = 0 + 4 = 4

Multiplication Properties of Integers:

1. Closure Property:  The product of two integers is always an Integers.

Example: Let two Integers are -3 and 5, then

-3 x 5 = -15 is an integers.

2. Commutative Property: Let a and b are two integers,

a x b = b x a

Example: Let two integers are -3 and -7 then

(-3) x (-7) = 21

(-7) x (-3) = 21

then (-3) x (-7) = (-7) x (-3)

Hence commutative property is verified

3. Associative Property: Let a, b and c are three integers, then

a x ( b x c) = (a x b ) x c

Example: Let three integers are -3, 5 and 6, then

-3 x ( 5 x 6 ) = -3 x 30 = -90

(-3 x 5 ) x 6 = -15 x 6 = -90

then -3 x ( 5 x 6 ) =  (-3 x 5 ) x 6

Hence Associative property is verified

4. Existence of Multiplicative Identity: For any integer a, we have

a x 1 = 1 x a = a

0 is called the additive identity.

Example: 5 is any whole number,

5 x 1 = 1 x 5 = 5

5. Distributive Property of Multiplication over Addition: Let a, b and c are three integers,

a x ( b + c ) = a x b + a x c

Example: Let 2, -3 and 4 are three integers,

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

2 x -3 + 2 x 4 = -6 + 8 = 2

Hence Distributive property is verified.

6. Distributive Property of Multiplication over Subtraction: Let a, b and c are three integers,

a x ( b - c ) = a x b - a x c

Example: Let 2, -3 and 4 are three integers,

2 x ( -3 - 4 ) = 2 x -7 = -14

2 x -3 + 2 x -4 = -6 + (-8) = -14

Hence Distributive property is verified.

Definition of Integers - Positive & Negative Integers | jmathslearning

noreply@blogger.com (Euclid_mathematics_classes) — Fri, 02 Jun 2017 13:17:00 +0000

Integers

An integer is a number that can be written without a fractional component that. It can be positive, negative or zero.It is denoted by "ℤ".

ℤ = { ....... –3, –2, –1, 0, 1, 2, 3, 4, 5, ........ }

Type of Integers

1. Positive Integers

2. Negative Integers

3. Non-negative Integers

1. Positive Integers

Whole numbers greater than zero are called Positive Integers. It is also called Natural Numbers and denoted by "ℤ⁺".

ℤ⁺^{= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ....... }}

2. Negative Integers

A negative integer is an integer to the left of zero on the number line. It is less than zero.It is denoted by "ℤ⁻".

ℤ⁻= { ........... ,−7, −6, −5, −4, −3, −2, −1}

3. Non-Negative Integers

An integer that is either 0 or positive. It is also called Whole Numbers and denoted by "ℤ^≠ ". Some Authors use symbol " ℤ^*".

ℤ^≠= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ....... }

We can understand Integers from a Number Line.

A number on the left is less than a number on the right.

we can understand with help of some examples.

3 is less than 5

-2 is less than 0

-5 is less than -2

A number on the right is greater than a number on the left.

we can understand with help of some examples.

11 is greater than 8.

0 is greater than -6.

-6 is greater than -11.

We can understand Integers from a Picture Diagram.

Properties of Whole numbers with examples

noreply@blogger.com (Euclid_mathematics_classes) — Thu, 01 Jun 2017 12:25:00 +0000

Properties of Whole Numbers

Addition Properties of Whole Numbers:

1. Closure Property: The sum of two whole numbers is always a whole numbers.

Example: Let two whole numbers 5 and 7 then

5 + 7 = 12 is a whole number.

2. Commutative Property: Let a and b are two whole numbers,

a + b = b + a

Example: Let two whole numbers are 5 and 6 then

5 + 6 = 11

6 + 5 = 11

then 5 + 6 = 6 + 5

Hence commutative property is verified
3. Associative Property: Let a, b and c are three whole numbers,
a + ( b + c) = (a + b ) + c
Example: Let three whole numbers are 3, 5 and 6 then
3 + ( 5 + 6 ) = 3 + 11 = 14
(3 + 5 ) + 6 = 8 + 6 = 14
then 3 + ( 5 + 6 ) = (3 + 5 ) + 6
Hence Associative property is verified
4. Existence of Additive Identity: For any whole number a , we have a + 0 = 0 +a = a
0 is called the additive identity.
Example: 4 is any whole number,
4 + 0 = 0 + 4 = 4

Multiplication Properties of Whole numbers:

1. Closure Property: The product of two whole numbers is always a whole numbers.

Example: Let two whole numbers 3 and 5 then

3 x 5 = 15 is a whole number.

2. Commutative Property: Let a and b are two whole numbers,

a x b = b x a

Example: Let two whole numbers are 3 and 7 then

3 x 7 = 21

7 x 3 = 21

then 3 x 7 = 7 x 3

Hence commutative property is verified
3. Associative Property: Let a, b and c are three whole numbers,
a x ( b x c) = (a x b ) x c
Example: Let three whole numbers are 3, 5 and 6 then
3 x ( 5 x 6 ) = 3 x 30 = 90
(3 x 5 ) x 6 = 15 x 6 = 90
then 3 x ( 5 x 6 ) = (3 x 5 ) x 6
Hence Associative property is verified
4. Existence of Multiplicative Identity: For any whole numbers a, we have a x 1 = 1 x a = a
0 is called the additive identity.
Example: 5 is any whole number,
5 x 1 = 1 x 5 = 5
5. Distributive Property of Multiplication over Addition: Let a, b and c are three whole numbers,

a x ( b + c ) = a x b + a x c

Example: Let 2, 3 and 4 are three whole numbers,

2 x ( 3 + 4 ) = 2 x 7 = 14

2 x 3 + 2 x 4 = 6 + 8 = 14

Hence Distributive property is verified.

What is whole numbers | Definition, Even & Odd Numbers

noreply@blogger.com (Euclid_mathematics_classes) — Thu, 01 Jun 2017 03:29:00 +0000

Whole Numbers

All natural numbers together with zero form the collection of all whole numbers. It is denoted by "W"

W = { 0,1,2,3,4,5,6,7,8,9,10, ............... }

Type of Whole Numbers

1. Even Numbers

2. Odd Numbers

1. Even Numbers

A number which is divisible by 2, it is called even number. It is denoted by "E".

E = { 0,2,4,6,8,10,12,14,16,18,20, ........ }

2. Odd Numbers

A number which is not divisible by 2, it is called odd number. It is denoted by "O".

O = { 1,3,5,7,9,11,13,,15,17,19,21, ....... }

We can understand whole numbers from a picture

Properties of natural numbers with examples

noreply@blogger.com (Euclid_mathematics_classes) — Wed, 31 May 2017 14:43:00 +0000

Properties of Natural Numbers

Addition Properties of Natural Numbers:

1. Closure Property: The sum of two natural numbers is always a natural numbers.

Example: Let two natural numbers 2 and 5 then

2 + 5 = 7 is a natural number.

2. Commutative Property: Let a and b are two natural numbers,

a + b = b + a

Example: Let two natural numbers are 5 and 7 then

5 + 7 = 12

7 + 5 = 12

then 5 + 7 = 7 + 5

Hence commutative property is verified
3. Associative Property: Let a, b and c are three natural numbers,
a + ( b + c) = (a + b ) + c
Example: Let three natural numbers are 3, 5 and 6 then
3 + ( 5 + 6 ) = 3 + 11 = 14
(3 + 5 ) + 6 = 8 + 6 = 14
then 3 + ( 5 + 6 ) = (3 + 5 ) + 6
Hence Associative property is verified
4. Existence of Additive Identity: For any natural number a , we have a + 0 = 0 +a = a
0 is called the additive identity.
Example: 4 is any natural number,
4 + 0 = 0 + 4 = 4

Multiplication Properties of Natural numbers:

1. Closure Property: The product of two natural numbers is always a natural numbers.

Example: Let two natural numbers 2 and 5 then

2 x 5 = 10 is a natural number.

2. Commutative Property: Let a and b are two natural numbers,

a x b = b x a

Example: Let two natural numbers are 5 and 7 then

5 x 7 = 35

7 x 5 = 35

then 5 x 7 = 7 x 5

Hence commutative property is verified
3. Associative Property: Let a, b and c are three natural numbers,
a x ( b x c) = (a x b ) x c
Example: Let three natural numbers are 3, 5 and 6 then
3 x ( 5 x 6 ) = 3 x 30 = 90
(3 x 5 ) x 6 = 15 x 6 = 90
then 3 + ( 5 + 6 ) = (3 + 5 ) + 6
Hence Associative property is verified
4. Existence of Multiplicative Identity: For any natural number a, we have a x 1 = 1 x a = a
0 is called the additive identity.
Example: 4 is any natural number,
4 x 1 = 1 x 4 = 4

What is natural numbers | Definition, Prime & Even Numbers

noreply@blogger.com (Euclid_mathematics_classes) — Tue, 30 May 2017 12:49:00 +0000

Natural Numbers

Counting numbers are known as natural numbers.

It is denoted by "N"

N = { 1,2,3,4,5,6,7,8,9,10,11,12,13, ....... }

Type of Natural numbers

1. Prime numbers

2. Composite numbers

1. Prime numbers

A number which have only two prime factor, it is called a prime numbers. It is denoted by "P".

P = { 2,3,5,7,11,13,17,19, ....... }

e.g. "2" have two factors 1 and itself .

Similarly some other examples written below :

3 -> 1 and 3

5 -> 1 and 5

7 -> 1 and 7

11 -> 1 and 11 etc.

2. Composite numbers

A numbers which have more than two factors, it is called a composite numbers.

{ 4,6,8,9,10,12,14,15,16,18, ...... }

e.g. "4" have more than two factors 1, 2 and 4

Similarly some other examples written below :

6 -> 1, 2, 3 and 6

8 -> 1, 2, 4 and 8

10 -> 1, 2, 5 and 10

12 -> 1, 2, 3, 4, 6 and 12 etc.

Note : " 1" is neither prime nor composite

We can understand numbers from a picture

UP Board Class 10 maths Paper set 822 (AV) solution 3 (a)

noreply@blogger.com (Euclid_mathematics_classes) — Mon, 27 Mar 2017 14:00:00 +0000

3(a). In English
Prove that cos π/8 + cos 3π/8 +cos 5π/8 +cos 7π/8 = 0
Solution: cos π/8 + cos 3π/8 +cos 5π/8 +cos 7π/8
= cos π/8 +cos 7π/8+ cos 3π/8 +cos 5π/8
= cos(π − 7π/8) +cos 7π/8+ cos 3π/8 +cos (π − 3π/8)
= − cos 7π/8 +cos 7π/8 + cos 3π/8 − cos 3π/8
= 0

3(a). In Hindi
fl) dhft, fd cos π/8 + cos 3π/8 +cos 5π/8 +cos 7π/8 = 0
Solution: cos π/8 + cos 3π/8 +cos 5π/8 +cos 7π/8
= cos π/8 +cos 7π/8+ cos 3π/8 +cos 5π/8
= cos(π − 7π/8) +cos 7π/8+ cos 3π/8 +cos (π − 3π/8)
= − cos 7π/8 +cos 7π/8 + cos 3π/8 − cos 3π/8
= 0

UP Board Class 10 maths Paper set 822 (AV) solution 2 (iv)

noreply@blogger.com (Euclid_mathematics_classes) — Mon, 27 Mar 2017 13:32:00 +0000

In English
2.(d) In the figure, if chord AC = 8 cm and chord BC = 6 cm find the radius of the circle with centre O .

Solution:

In Hindi
2 (d). Fp= esa] O dsUnz okys o`Rr dh thok AC = 8 lseh rFkk thok BC = 6 lseh gks] rks o`Rr dh f=T;k Kkr dhft,A

Solution:

UP Board Class 10 maths Paper set 822 (AV) solution 2 (iii)

noreply@blogger.com (Euclid_mathematics_classes) — Sun, 26 Mar 2017 04:56:00 +0000

2 (c) If 14 is the median of numbers 7, 8, (2 x - 3), (2 x - 1), 15 17, 20, and 22 arranged in ascending order then find the value of x.
Solution: In English

In Hindi

;fn vkjksgh dze esa la[;kvksa 7, 8, (2 x - 3), (2 x - 1), 15 17, 20 vkSj 22 dh ekf/;dk 14 gS] rks x. dk eku Kkr dhft,S

Solution:

UP Board Class 10 maths Paper set 822 (AV) solution 2 (ii)

noreply@blogger.com (Euclid_mathematics_classes) — Sun, 26 Mar 2017 04:22:00 +0000

2(ii) In English
A purchaser gets a rebate of 12.5% on television of the value Rs. 9000. After rebate, 10% commercial tax is charged on remaining amount, Then find the amount of commercial tax.
Solution: Given
Market price of a television = Rs. 9000
Discount = 12.5%
Cost price of a television = MP[1 − (D/100)]
= 9000 ✖ [1 − (12.5/100)]
= 9000 ✖ [ 1 − 0.125]
= 9000 ✖ 0.825
= Rs. 7875
Rate of sales Tax = 10%
Sales Tax = 10 % of 7875
= Rs 787.5
In Hindi
,d dzsrk dks 9]000 : ewY; ds Vsyhfotu ij 12.5% dh NwV feyrh gSA NwV ds ckn “ks’k jkf”k ij 10% - O;kikj dj yxrk gS] rks O;kikj dj dh jkf”k Kkr dhft,A
Solution: ज्ञात है
टेलीविज़न का अंकित मूल्य = 9000 :
छूट = 12.5%
टेलीविज़न का मूल्य = MP[1 − (D/100)]
= 9000 ✖ [1 − (12.5/100)]
= 9000 ✖ [ 1 − 0.125]
= 9000 ✖ 0.825
= Rs. 7875
व्यापार कर के दर = 10%
व्यापार कर = 10 % of 7875
= Rs 787.5

JMATHSLEARNING

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.

Theorem : The sum of the angles of a triangle is 180°

Median, Altitude, Incentre and Circumcircle

Median

Altitudes

Incentre of a Triangle

Circumference of a Triangle

Triangle definition | Equilateral Isosceles Scalene and Acute angled Right angle Obtuse angled Triangle

Triangle

Type of Triangles:

1. On the basis of sides:

2. On the basis of angles:

What is angle | Type of angles

Angle

Two rays starting from a common point form an angle. In ∠ AOB, O is the vertex, OA and OB are the two rays.

Measure of a angle

Types of angles

Bisector of an angle

Related Angles

Adjacent angles

Linear Pair of Angle

Angle at a point

Vertically opposite angles

Parallel lines

Transversal

The Angles formed when a transversal cuts two lines

Various Type of Numbers Chart

Definition | Point, Ray, Line, Collinear Points, Plane, Parallel Lines

Point

Line Segment

Ray

Line

Incidence Axioms on lines

Collinear Points

Intersecting lines

Concurrent Lines

Plane

Parallel Lines

Euclid's Axioms & Postulates | examples

History

A Few the Euclid’s Definitions :

Basic Geometrical Concepts:

Euclid’s Axioms:

Euclid’s Five Postulates:

Algebraic Identites

Formulae for binomial

1. (a + b)2 = a2 + 2ab + b2

2. (a – b)2 = a2 – 2ab + b2

3. a2 – b2 = (a + b) . (a – b)

4. (a + b)3 = a3 + b3 + 3ab (a + b) = a3 + b3 + 3a2b + 3ab2

5. (a – b)3 = a3 – b3 – 3ab (a – b) = a3 – b3 – 3a2b + 3ab2

6. (a3 + b3) = (a + b)(a2 – ab + b2)

7. (a3 – b3) = (a – b)(a2 + ab + b2)

formula for Trinomial

8. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

9. (a3 + b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

10. If (a + b + c) = 0 ⇒ a3 + b3 + c3 = 3abc

Remainder Theorem | Factor Theorem

Remainder Theorem

Factor Theorem

What is Polynomial | Various kinds of Polynomials

Constant

Variable

Algebraic Expression

Terms of an Algebraic Expression

Polynomial

Degree of Polynomial in one variable

Degree of a Polynomial in two or more variables

Various Type of Polynomials

1. Polynomials of Various Degrees

(i). linear polynomial

(ii). Quadratic Polynomial

(iii). Cubic Polynomial

(iv). Biquadratic Polynomial

2. Number of terms in a Polynomial:

(i). Monomial

(ii). Binomial

(iii). Trinomial

Constant polynomial

1. (a + b)² = a² + 2ab + b²

2. (a – b)² = a² – 2ab + b²

3. a² – b² = (a + b) . (a – b)

4. (a + b)³ = a³ + b³ + 3ab (a + b) = a³ + b³+ 3a²b + 3ab²

5. (a – b)³ = a³ – b³ – 3ab (a – b) = a³ – b³– 3a²b + 3ab²

6. (a³ + b³) = (a + b)(a² – ab + b²)

7. (a³ – b³) = (a – b)(a² + ab + b²)

8. (a + b + c)² = a²+ b² + c² + 2ab + 2bc + 2ca

9. (a³ + b³ + c³ – 3abc) = (a + b + c)(a²+ b² + c² – ab – bc – ca)

10. If (a + b + c) = 0 ⇒ a³+ b³+ c³= 3abc