Current Affairs 5th Class

Geometry

Category : 5th Class

GEOMETRY

 

FUNDAMENTALS         

  •                   In geometry, there are three basic terms point, line and plane.
  •                   Point: A point does not have length, breadth and height. It is a mark of position and is represented by a dot.
  •                   Line: A line normally refers to a straight line which extends indefinitely in both the directions. Thus, it has length but no breadth and no height.

Example: If you hold a thread taut between two hands, it represents part of a line.

  •                  Plane: A plane has two dimensions, length and breadth, but no height.

Example: A piece of paper represents a plane, Top of a table represents plane, etc.

Passing through a point, an infinite number of lines can be drawn.

\[{{l}_{1,}}{{l}_{2.................}}{{l}_{n}}\] All pass through ?P?

These lines are also called CONCURRENT lines and the point P is called point of concurrence.

  •                  Two lines in a place are either intersecting or parallel

 

      Collinearity of Points

Three points A, B, C in a place are collinear if they lie on the same straight line.

  •                   One another way of testing collinearit6y of three points A, B and C is AB + BC = AC

If this equality holds, then points are collinear.

If this equality doesn?t hold, then points are non-collinear.

  •                   Ray: Part of line which extends indefinitely from a given point ?P? is called a ray.

\[{{l}_{1,}}{{l}_{2,}}{{l}_{3..........}}{{l}_{n}}\]are all rays.

 

     Line Segment 

  •                 Part of the line between two given points A and B on the line, is called a line segment.

  •                 Line segment AB is represented as\[\overline{AB}\]. It is measured in ?cm? or ?inch?
  •                Two line segments AB and CD are equal, they are of same length.

Example: if \[\overline{AB}\]=10cm and \[\overline{CD}=4\]inch then \[\overline{AB}=\overline{CD}\](because 1 inch=2.5 cm)

 

ANGLE

  •                An angle is a figure formed by two rays with same initial point

Example:  

Rays \[{{l}_{1}}\] and \[{{l}_{2}}\] form an angle between them; this angle is represented as \[\theta =\angle POQ.\angle POQ\] can simply be written as\[\angle O\].

 

  •                Unit of measurement of angle is degrees, which is represented as\[^{o}\]\[(eg:\,\,{{30}^{o}},\,\,{{60}^{o}},\,\,{{90}^{o}}\,\,etc)\]

 

      Angles in a triangle

A plane figure bounded by three line segment is called a triangle. 

Example: 

It has three angles \[\angle BAC\] (also called\[\angle A\]), \[\angle ABC\] (also called\[\angle B\]) and\[\angle ~ACB\] (also called\[\angle C\]).

  •                 Sum of angles of\[\Delta \]is always\[{{180}^{o}}\].
  •                 Right- angled \[\Delta \] or a right- angle\[\Delta \]:

In a\[\Delta \], if one angle \[={{90}^{o}}\]then it is called right angled triangle.

ABC is a right \[\Delta \] in which \[\angle B={{90}^{o}}\]

In a right-angle\[\Delta \], side opposite to right \[\angle \]is called hypotenuse. Other two sides are called base and perpendicular. The relation between these sides is given by Pythagoras as

\[{{\text{(Base)}}^{\text{2}}}\text{+(Perpendicular}{{\text{)}}^{\text{2}}}\text{=(Hypotenuse}{{\text{)}}^{\text{2}}}\]\[A{{B}^{2}}+B{{C}^{2}}=A{{C}^{2}}\] (in right \[\Delta ABC\])

 

  •                       Other types of \[\Delta \] are:

(i) Isosceles \[\Delta \](a \[\Delta \] in which two sides are equal)

(ii) Equilateral \[\Delta \](a \[\Delta \] in which all three sides are equal)

(iii) Scalene triangle \[\Delta \] (a \[\Delta \] in which no side is equal)

 If in \[\Delta \] ABC, AB=AC, then it is isosceles, also \[\angle \]B=\[\angle \]C

 If  in equilateral\[\Delta ABC\],

\[AB=BC=CA\], also\[\angle A=\angle B=\angle C={{60}^{o}}\]

 

QUADRILATERALS

  •                    A plane figure (meaning a figure drawn in a plane) bounded by four line segment is called a quadrilateral.

 

Type of Quadrilaterals are as follows:

(i) Trapezium: A quadrilateral having only one pair of parallel sides

(ii) Isosceles trapezium: It is a special type of trapezium in which non-parallel sides are equal i. e. \[AB||CD\] and\[AD=BC\]

(iii) Parallelogram: A quadrilateral having both pairs of opposite sides are parallel i. e.,\[AB||CD\]and\[AD||BC\]

As a natural consequence of this, AB = CD and AD = BC, and \[\angle A=\angle C\]and\[\angle B=\angle D\]

(iv) Rhombus: It is a special type of parallelogram in which all side are equal

\[AB||CD\]

\[AD||BC\]

\[AB=BC=CD=DA\]

Also, \[\text{ A}C\bot BD\](i.e. diagonals are perpendicular to each other)

And,\[AO=CO\] and \[BO=DO\] (i.e., diagonals bisect each other)

  •                   Rectangle: A parallelogram who?s each angle is right angle.

\[AB||CD;\,\,AD||BC\]

\[AB=CD;\,\,AD=BC\]

\[\angle A=\angle B=\angle C=\angle D={{90}^{o}}\]

  •                   Square: A special type of rectangle whose all sides are equal.

\[AB||CD;\,\,AD||BC\]

\[AB=BC=CD=DA\]

\[\angle A=\angle B=\angle C=\angle D={{90}^{o}}\]

  •                   Kite: A quadrilateral which has equal adjacent sides but unequal opposite sides.

Adjacent Sides: \[AB=BC\] and\[AD=CD\]

Opposite sides: \[AB\ne CD\]; \[BC\ne AD\]

 

CIRCLE

  •                    A circle is a set of points in a plane whose distance from a fixed point is constant

'O' is the fixed point called center.

'P' is movable point.

OP is called radius of a circle

  •                   Diameter: A line segment passing through center and having its end points on the circle.

\[\therefore \] AB or CD are diameters.

Since infinite line segments can be drawn through O, therefore, numbers of diameters are infinite.

Now, look at the figure below:

AB = diameter

  •                 EF, which meets circle at two points is called chord of circle.
  •                 Diameter is the largest chord.
  •                Line through PQ where P and Q are points on circle, is called secant of a circle
  •                 Line, (through GH) which touches circle at only one point is called Tangent to the circle.

 



LIMITED OFFER HURRY UP! OFFER AVAILABLE ON ALL MATERIAL TILL TODAY ONLY!

You need to login to perform this action.
You will be redirected in 3 sec spinner

Free
Videos