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CIRCLE

This content is part of NCERT Solutions for Class 10 Maths Chapter 10 Circles

           A circle is the locus of a points which moves in a plane in such a way that its distance from a fixed point

 remains constant.

 9.2        SECANT AND TANGENT :

                  Secant to a circle is a line which intersects the circle in two distinct points.

                  A tangent to a circle is a line that intersects the circle in exactly one point.

 9.3        THEOREM :

Statement :       A tangent to a circle i perpendicular to the radius through the point of contact.

Given :             A circle C (O, r) and a tangent AB at a point P.

 To prove :        OP AB

 Construction :  Take any points Q, other than P on the tangent AB. Join OQ. Suppose OQ meets the circle at R.

 Proof:              Among all line segments joining the point O to a point on AB, the shorted one is perpendicular to

 AB. So, to prove that OP AB, it is sufficient to prove that OP is shorter than any other segment

 joining O to any point of AB.

Clearly  OP = OR

Now,     OQ  OR + RQ

      OQ > OR

      OQ > OP (OP = OR)

This content is part of NCERT Solutions for Class 10 Science Chapter 3 Metals and Non-metals

GENERAL PROPERTIES OF METALS AND NON-METALS :

 7.3 (a) Electronic Configuration of Metals :

 The atoms of metals have 1 to 3 electrons in their outermost shells. For example, all the alkali metals have one electron in their outermost shells (lithium 2, 1; sodium 2,8,1: potassium 2,8,8,1 etc.)

Sodium, magnesium and aluminum are metals having 1,2 and 3 electrons respectively in their valence shells. Similarly, other metals have 1 to 3 electron in their outermost shells.

ANGLE OF ELEVATION :

This content is part of NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry

           In order to see an object which is at a higher level compared to the ground level we are to look up. The line joining the object and the eye of the observer is known as the line sight and the angle which this line of sight makes with the horizontal drawn through the eye of the observer is known as the angle of elevation. Therefore, the angle of elevation of an object helps in finding out its height (figure)

 12.2      ANGLE OF DEPRESSION :

             When the object is at a lower level tan the observer’s eyes, he has to look downwards to have a view of the object. It that case, the angle which the line of sight makes with the horizontal thought the observer’s eye is known as the angle of depression (Figure).

             ILLUSTRACTIONS :

             In order to see an object which is at a higher level compared to the ground level we are to look up. The line joining the object and the eye of the observer is known as the line sight and the angle which this line of sight makes with the horizontal drawn through the eye of the observer is known as the angle of elevation. Therefore, the angle of elevation of an object helps in finding out its height (figure)

 12.2      ANGLE OF DEPRESSION :

             When the object is at a lower level tan the observer’s eyes, he has to look downwards to have a view of the object. It that case, the angle which the line of sight makes with the horizontal thought the observer’s eye is known as the angle of depression (Figure).

CONGRUENT AND SIMILAR FIGURES:

This content is part of NCERT Solutions for Class 10 Maths Chapter 6 Triangles

Two geometric figures having the same shape and size are known as congruent figures. Geometric figures having the same shape but different sizes are known as similar figures.

8.2       SIMILAR TRIANGLES:

               Two triangles ABC and DEF are said to be similar if their

               (i)            Corresponding angles are equal.

                               i.e. ÐA = ÐD, ÐB = ÐE, ÐC = ÐF

                               And,

MENSRTION:

This content is part of NCERT Solutions for Class 10 Maths Chapter 13 Surface Areas and Volumes

           Figure lying in a plane is called a plane figure. A plane figure made up of lines or curve or both, is said to be a closed figure if it has on free ends. Closed figure in a plane covers some part of the plane, then magnitude o that part of the plane is called the area of that closed figure. the unit of measurement of that part of the plane is called the area of that closed figure. the unit o measurement of area is square unit (i.e. square centimeter, square metre etc.)

  13.1    (a)Mensuration of a Triangle:

             perimeter = a + b + c

           Area = × Base × Height

                                                  Heron’s formula:

             Area =

This content is part of NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

RECTANGULAR CO-ORDINATES :

Take two perpendicular lines X’OX and Y’OY intersecting at the point O. X’OX and Y’OY are called the co-ordinate axes. X’Ox is called the X-axis, Y’OY is called the Y-axis and O is called the origin. Lines X’OX and Y’OY are sometimes also called rectangular axes.

7.1 (a) Co-ordinates of a Point :

PROGRESSIONS :

This content is part of NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions

Those sequence whose terms follow certain patterns are called progression. Generally there are three types of progression.

               (i) Arithmetic Progression (A.P.)  (ii) Geometric Progression (G.P.)  (iii) Harmonic Progression (H.P.)

 6.2       ARTHMETIC PROGRESSION :

A sequence is called an A.P., if the difference of a term and the previous term is always same. i.e. d = tn+1 – tn = Constant for all . The constant difference, generally denoted by ‘d’ is called the common difference.

 Ex.1        Find the common difference of the following A.P. : 1,4,7,10,13,16 ......

Sol.         4 - 1 = 7 - 4 = 10 - 7 = 13 - 10 = 16 - 13 = 3 (constant).