# 5th Class Mathematics Geometry

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$\angle A=\angle B=\angle C={{60}^{o}}$

•                    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.

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